Questão 8 (Física)

**Clarifying dynamic concepts** I'm diving into the topic of uniform helical speed versus nonuniform velocity. I'm trying to justify this with the idea that the force is perpendicular, which makes things interesting because it doesn't change the magnitude. There are some energy considerations, too. I wonder if having a condition where cosθ equals an integer inverse is necessary, but it's definitely a nice boundary condition to think about in physics. It’s all adding layers to my understanding.

**Exploring sine approximation** I'm wondering if the sine approximation is only relevant when the calculation involves time. They provide sinθ as approximately θ to help solve angular relations. Maybe when a particle follows a helix, it "returns to the same line" with each turn. The position after one full revolution depends on v cos θ rather than sin θ. I think I could derive a condition on cos θ using speed v, without needing the small-angle assumption.**Exploring cosine conditions** I'm looking into the condition that cosθ must satisfy, which involves some integer n. After simplifying, I find that cosθ = qBL/(2πn√(2mqV)). I checked the dimensions and confirmed they align. Cosθ must be less than or equal to 1, and if n=1, that’s the first return. There might also be a pitch condition to consider. It seems crucial to clarify if they really mean the first return with n=1, so I'll explain n as the number of cyclotron periods.**Clarifying recurrence expectations** I'm considering if the wording around "um desses retornos" suggests that they expect a cycle. In physics, standard recurrence indicates that it passes through the same field line again, which mathematically means n cycles. I want to be careful not to assume it’s just the first cycle. I'll present a general case while still noting that the first one is a special instance. It's important to clarify these points to avoid any confusion!**Considering final answer format** I think I should interpret point2 as the point reached in the first return, based on the phrase "em um desses retornos.” I need to state both conditions concisely. The final line should be specific but also include n. I can format the answer to say c) \(\cos\theta=...\), \(n\in \mathbb N^+\) (if n=1). I want to ensure the dimensions are correct and define B as the magnitude. Let's focus on delivering this clearly.

For part a, I find v from energy conservation: qV equals half mv², giving v = √(2qV/m). For part b, I break the velocity into components parallel and perpendicular to B — the parallel part stays constant while the perpendicular part undergoes circular motion with radius mv sinθ/(qB) and period 2πm/(qB), so together they trace a helix with pitch equal to v cosθ times that period. For part c, the particle returns to the same field line after each full period of that circular motion. Reaching point 2 after n periods requires L = n·v cosθ·T, which lets me solve for cosθ in terms of L, B, n, and the other constants: cosθ = (BL/(2πn))√(q/(2mV)), with n a positive integer satisfying n ≥ BL√(q/(2mV))/(2π) to keep cosθ ≤ 1.

Using the standard speed from acceleration, v = sqrt(2qV/m), I'm working through the helical motion with period T = 2πm/(qB) and pitch p = v cosθ T. For the particle to pass through the second point, the distance L must equal n times the pitch for some positive integer n, giving cosθ = (LB/(2πn))√(q/(2mV)), with the constraint that cosθ ≤ 1.

The problem originates from ITA 2027, Physics, second phase. The problem involves a charged particle entering a uniform magnetic field at an angle θ. It returns to the same field line after passing point 2 at distance L from point 1. I need to find the entry velocity and identify the type of motion inside the field. The entry velocity is v = sqrt(2 q V / m). The third part of the problem asks for the condition on cos θ so the particle passes through point 2. Inside the uniform magnetic field, the particle performs helical motion. The component of velocity parallel to the field remains constant, while the perpendicular component causes circular motion around the field line. This spiral trajectory explains why the particle returns to the same field line after passing point 2. The period of the cyclotron motion is T = 2π m / (q B). The pitch of the helix equals the parallel velocity multiplied by this period. This distance along the field line between returns is independent of the total velocity. The distance between returns along the field line is L = n·v cos θ·2π m/(q B), where n is a positive integer. Substituting the entry velocity v = sqrt(2 q V / m) yields the condition on cos θ in terms of the given quantities. The circular orbit intersects the original field line only at the starting point. The particle therefore does not return to the same field line after each revolution. This contradicts the problem's assumption and invalidates the helical motion model. The circular orbit intersects the original field line only at the starting point. After each complete revolution, the particle returns to that same point in the xy plane and therefore to the same field line. The earlier concern about a contradiction is resolved. The condition on cos θ is derived from the pitch of the helix. The distance between consecutive returns along the field line is p = v cos θ·2π m/(q B). Because point 2 lies on the same field line at distance L, we require L = n·p for some positive integer n. Substituting v = sqrt(2 q V / m) yields cos θ = (q B L)/(2 π n m sqrt(2 q V / m)). The condition on cos θ simplifies to cos θ = (B L)/(2π n)·√(q/(2 m V)), where n is a positive integer. The condition can also be written as n = (q B L)/(2 π m v cos θ), where n must be a positive integer. Since cos θ is positive, this implies that the ratio (q B L)/(2 π m v) divided by cos θ must be a natural number. The condition on cos θ should be expressed in terms of the given quantities. The small-angle approximation is likely unnecessary for this problem.

We need answer in Portuguese. Need solve physics question. Let's restate. Problem: Charged particles accelerated from rest by potential difference V>0. Then enter vacuum region where only uniform magnetic field B acts. Particle mass m charge q>0 enters region at point 1, velocity v forming angle theta with direction of magnetic field. Figure: B field horizontal to right (lines arrows), particle enters point 1, velocity vector downward-right making angle theta with field direction (horizontal right). Then trajectory helical? It says during trajectory, particle returns to pass through same field line on which motion began. In one of these returns, it passes through point 2, located distance L from point 1. Determine a) speed magnitude upon entering magnetic field; b) type of motion inside magnetic field, justify; c) condition that cos theta must satisfy for particle to pass through point 2. Express in terms of data. Need solve. Let's analyze. Part a: accelerated by potential difference V: qV = (1/2) m v^2 => v = sqrt(2qV/m). Nonrelativistic. Part b: Motion in uniform magnetic field: velocity has component parallel to B (v_parallel = v cos theta) and perpendicular (v_perp = v sin theta). Magnetic force q v × B acts only on perpendicular component, causing circular motion in plane perpendicular to B with radius r = m v_perp / (qB). Parallel component constant. Thus trajectory is a helix (helicoidal) with axis along B, uniform circular motion in transverse plane plus uniform translation along B. So type: movimento helicoidal uniforme (hélice cilíndrica) / composição de movimento retilíneo uniforme na direção de B e movimento circular uniforme no plano perpendicular a B. Justify: força magnética perpendicular à velocidade e a B; componente paralela não sofre força, componente perpendicular realiza MCU; raio etc. Part c: Condition cos theta for particle passes through point 2. Let's understand geometry. Particle enters at point 1 on a field line? "volta a passar pela mesma linha de campo na qual o movimento teve início." The magnetic field lines are horizontal straight lines. Point 1 is on a field line (the line through point 1). After entering, it follows helix. It returns to same magnetic field line (same line of B?) maybe after one or more full circular revolutions, because perpendicular motion circular; when it completes an integer number of turns, its transverse position returns to initial transverse coordinate, so it crosses the same field line (line parallel to B through initial point). "Em um desses retornos, ela passa pelo ponto 2" In one of these returns, it passes through point 2, located at distance L from point 1. Point 2 lies on same line of field as point 1? Figure: point 2 is on same horizontal line as point 1, distance L to right. Yes "mesma linha de campo na qual o movimento teve início" and point 2 situated at distance L from point 1, on that line. Thus for particle to pass through point 2, after some number n of complete circular revolutions, it has translated along B by distance L? Actually point 2 on same field line, so transverse displacement zero, longitudinal displacement along B = L. The time to complete n cyclotron periods T_c = 2π m / (qB) each. During one period, displacement along B = v_parallel * T_c = (v cos theta) * (2π m / qB). For n turns, displacement = n * v cos theta * (2π m / qB) = L. Thus cos theta = L q B / (2π n m v). But need maybe condition? They ask "a condição que cos θ deve satisfazer para que a partícula passe pelo ponto 2. Expresse sua resposta em função dos dados do problema." Could be cos theta = L q B / (2π n m v) for some positive integer n? Or maybe because point 2 is "um desses retornos" but distance L may correspond to first return? Let's inspect. They might want condition for cosθ? Let's derive carefully. When particle returns to same field line, the transverse displacement is zero, i.e., after an integer number of turns. The point 2 is at distance L from point 1 along field line. The displacement along B after n complete turns equals L. But also note: The particle may pass through point 2 after multiple turns? "Em um desses retornos, ela passa pelo ponto 2" means there are multiple returns; one of them happens at point 2. Since returns occur at discrete positions x_n = n * v_parallel * T (assuming start at x=0). So L must equal n * v cosθ * T for some positive integer n. Therefore cosθ = L q B / (2π n m v). Since v = sqrt(2qV/m), perhaps express in terms of V: cosθ = L q B / (2π n m sqrt(2qV/m)) = L q B / (2π n sqrt(2 m q V))? Let's compute: m v = m sqrt(2qV/m)= sqrt(2 m q V). So cosθ = L q B / (2π n sqrt(2 m q V)) = L B / (2π n) * q / sqrt(2 m q V)? = L B sqrt(q) / (2π n sqrt(2 m V))? But maybe keep in terms m,q,V,B,L,n. Also maybe they expect cosθ = L B sqrt(q)/(2π n sqrt(2mV))? We can simplify: q / sqrt(2 m q V)= sqrt(q^2/(2 m q V)) = sqrt(q/(2 m V)). Yes = sqrt(q/(2 m V)). Thus cosθ = (L B / (2π n)) * sqrt(q/(2 m V)) = L B sqrt(q)/(2π n sqrt(2 m V)). Wait multiply: L B/(2π n) * sqrt(q/(2mV)) = L B sqrt(q) / (2π n sqrt(2mV)). Yes. But maybe they don't want n? Let's see question c: "a condição que cos θ deve satisfazer para que a partícula passe pelo ponto 2. Expresse sua resposta em função dos dados do problema." It might be that "passe pelo ponto 2" only if after some number of complete revolutions, the displacement along B equals L. Thus condition involves an integer n? "condição" could be an equation with integer n, where n=1,2,...; maybe because there are many possible? But maybe due to geometry, if it passes through point 2, then L = n * pitch? But "condição que cosθ deve satisfazer" maybe they want cosθ = L q B / (2π m v) (n=1)? Let's examine wording: "Em um desses retornos, ela passa pelo ponto 2, situado a uma distância L do ponto 1." It doesn't say first return; "em um desses retornos" could imply one of the returns. Then it passes point 2 at one return. So yes n can be any positive integer. The condition: cosθ = L B q / (2π n m v) with n = 1,2,3,... Maybe they might define n as number of complete turns. But maybe because "passe pelo ponto 2" first time? The figure shows point 2 on the line; after first return (one full turn) maybe at distance equal to pitch; but if point 2 is arbitrary L, then if L equals pitch, particle passes after 1 turn; if L equals 2 pitch, after 2 turns, etc. The condition should include integer. But are there other possibilities? The particle could pass through point 2 not exactly at a return? It says "volta a passar pela mesma linha de campo na qual o movimento teve início. Em um desses retornos, ela passa pelo ponto 2" So point 2 is on that line, and at one return it passes there. It doesn't necessarily say it only passes at returns? But if it is on same field line, yes it passes through that line only at returns? Actually after entering at point 1, the particle may cross the original field line more than once if its transverse motion? Let's visualize: magnetic field lines are parallel horizontal; the "same field line" is the line through point 1 parallel to B. The particle's trajectory is helix; its projection onto transverse plane is circle. It crosses the original field line whenever its perpendicular displacement is zero, i.e., after each full period (and initial time? At t=0 at origin). Actually a charged particle in uniform B with initial velocity at angle to B moves in a helix. In the plane perpendicular to B, it performs a circle through the starting point (because initial perpendicular velocity radial? Wait: The guiding center drifts? Let's derive: v_perp rotates; position perpendicular is r_perp(t) = (v_perp/ω)(sin(ωt), 1 - cos(ωt)) if initial velocity along one axis? Starting at origin, after one full period returns to origin in transverse plane. Yes, crosses same field line at t = nT. Good. So no other crossings. Thus L must be n times pitch. But maybe point 2 lies on the original field line after one return, not multiple? The problem statement: "Em um desses retornos, ela passa pelo ponto 2" could be any. Need include n. Let's derive c fully. Let v = sqrt(2qV/m). Component parallel: v_∥ = v cosθ. Magnetic force no work, so speed constant. Perpendicular component v_⊥ = v sinθ. Cyclotron angular frequency ω = qB/m. Period T = 2π m/(qB). Pitch (distance advanced along B in one period): p = v_∥ T = (v cosθ)(2π m / qB). After n periods (n=1,2,...), displacement along B = n p = L. Then cosθ = L / (n v T) = L q B / (2π n m v). Since v = sqrt(2qV/m), cosθ = L q B / (2π n m sqrt(2qV/m)) = L q B / (2π n sqrt(2m q V)) = L B sqrt(q) / (2π n sqrt(2m V))? Let's verify: q / sqrt(2m q V) = sqrt(q^2/(2m q V)) = sqrt(q/(2mV)). Yes. So cosθ = L B / (2π n) * sqrt(q/(2mV)) = L B sqrt(q) / (2π n sqrt(2mV)). Also can write = L B /(2π n) sqrt(q/(2mV)). Maybe leave as L q B / (2π n sqrt(2 m q V)). Which is best? The data include m, q, V, B, L. So expression in terms of dados: cosθ = L q B / (2π n sqrt(2 m q V)) or with n. But maybe if no n, take n=1? Let's see if "a condição que cosθ deve satisfazer" perhaps because if L is arbitrary, the condition is that there exists a positive integer n satisfying; but "condição" may be equality with n? They ask "a condição que cos θ deve satisfazer para que a partícula passe pelo ponto 2." Since L is fixed, it's a quantization condition on cosθ: cosθ = (L q B)/(2π n sqrt(...)) for some integer n. But perhaps they expect solving with "n" as an integer? Let's check if typical ITA problem? This might be from ITA 2024? There may be official solution. Let's recall maybe. Question: "Observa-se que, durante a trajetória, a partícula volta a passar pela mesma linha de campo na qual o movimento teve início. Em um desses retornos, ela passa pelo ponto 2, situado a uma distância L do ponto 1. Desprezando efeitos relativísticos, determine ... c) a condição que cos θ deve satisfazer para que a partícula passe pelo ponto 2. Expresse sua resposta em função dos dados do problema." They might want: cosθ = L q B / (2π n m v), n=1,2,3... But maybe they want "cosθ = n L q B/(2π m v)"? Wait rearrange: If L = n p, then cosθ = L/(n vT). So n in denominator. But if they solve for cosθ perhaps they might define n as number of turns; yes. Let's see if there is an alternative: Maybe "passe pelo ponto 2" after first return "em um desses retornos" but if point 2 is at distance L, condition might be that distance traveled along field line in one period equals L, so L = v cosθ * 2π m/(qB) => cosθ = L q B/(2π m v). Since they didn't mention n? But they said "em um desses retornos" could mean "at one of these returns", so after some number of turns; but perhaps if it's at point 2, which is on the same line, and "em um desses retornos" not "no primeiro retorno". Hmm. Let's parse original Portuguese: "Observa-se que, durante a trajetória, a partícula volta a passar pela mesma linha de campo na qual o movimento teve início. Em um desses retornos, ela passa pelo ponto 2, situado a uma distância L do ponto 1." It says "Em um desses retornos" = "In one of these returns", not "no primeiro". So it definitely acknowledges multiple returns. Thus integer n. But maybe they want condition on cosθ involving an arbitrary integer n? Could be. Let's examine part b: "o tipo de movimento realizado pela partícula no interior da região de campo magnético, justificando sua resposta". Answer: movimento helicoidal uniforme (hélice cilíndrica), composição de um movimento circular uniforme no plano perpendicular ao campo com um movimento retilíneo uniforme na direção do campo. Justify: A força magnética é perpendicular à velocidade e ao campo; ela não realiza trabalho, então o módulo da velocidade é constante; a componente da velocidade paralela a B permanece constante (não há força nessa direção); a componente perpendicular descreve MCU de raio R = m v senθ/(qB). Thus helix. Part a: v = sqrt(2qV/m). Part c: condition as above. Need ensure notation: use θ (teta), possibly "senθ" etc. Use "≈" maybe not needed. The "Quando necessário, use ... Para θ << 1 assuma senθ ≈ θ." Not needed? Part c might involve cos theta not small? Probably not. Let's derive c more rigorously maybe include geometric series? Let's formulate. At t=0, particle at point 1. Velocity components: along B: v cosθ; perpendicular: v sinθ. Magnetic field B along positive x-axis. Let's choose coordinates: x along B, y,z transverse. Force q v × B. The motion: x(t) = v cosθ t. Transverse: circular motion with radius R = m v sinθ/(qB), angular frequency ω = qB/m. Starting at origin with velocity maybe along y-direction? It will trace circle; after time T = 2π/ω, it returns to transverse position (0,0). Thus at times t_n = nT, its coordinates are x_n = v cosθ * nT = n * (2π m v cosθ)/(qB), y=0,z=0. These are "retornos" to original field line. Point 2 is at distance L from point 1 along B, so its coordinate x = L. Thus L = n * (2π m v cosθ)/(qB). Solve for cosθ: cosθ = L q B / (2π n m v). Since v = sqrt(2qV/m), yes. Could also express using kinetic energy: m v = sqrt(2m qV). Then denominator 2π n sqrt(2m q V). Good. Potential issue: The speed v in part a is sqrt(2qV/m). If q positive, acceleration from rest through potential difference V: work qV. Yes. Let's include units? Not necessary. Let's think if "condição que cos θ deve satisfazer" could be "cosθ = L q B/(2π m v)" and also "com n inteiro positivo"? Maybe answer: cosθ = L q B/(2π n m v), n=1,2,3,... But maybe they expect \( \cos\theta = \frac{L q B}{2\pi m v} \) because "em um desses retornos" first return? Let's check statement: "Em um desses retornos, ela passa pelo ponto 2, situado a uma distância L do ponto 1." If it passes at one of returns, then distance from point 1 to point 2 is L. If it passes at n-th return, L = n * pitch. They may want condition "cosθ = L B q/(2π n m v)" with n natural? But the problem might consider point 2 as the first point after point 1 on same field line reached at a return? But they didn't say "primeiro retorno". Let's search memory: This seems from ITA 2024/2025? The figure and statement maybe from "ITA 2024 Física" question? I vaguely recall an ITA problem about mass spectrometer: accelerated by V, enter magnetic field at angle theta; "A partícula volta a passar pela mesma linha de campo na qual o movimento teve início. Em um desses retornos, ela passa pelo ponto 2, situado a uma distância L do ponto 1. Determine ... c) a condição que cos θ deve satisfazer..." Maybe official answer might be \( \cos\theta = \frac{L q B}{2\pi m v} \) perhaps because they consider the first return? Let's see. If they wanted integer multiple, the condition would have "n" and maybe they'd say "para que a partícula passe pelo ponto 2, é necessário que \( \cos\theta = \frac{L q B}{2\pi n m v}\), com n=1,2,3,...". But can a "condition" be multiple possibilities? Yes. However, in a mass spectrometer, maybe they select velocity? The angle theta is determined by geometry? Hmm. Let's analyze: Particle enters at point 1, with speed v and angle theta. It follows helix. It will return to original field line at distances \(s_n = n L_0\), where \(L_0 = v \cos\theta T\). The point 2 is at a fixed distance L from point 1. If L is arbitrary, in order for particle to pass through point 2, L must be an integer multiple of the pitch \(L_0\). But since L and v are given, cosθ must satisfy relation. So there may be multiple possible cosθ (for different n). But perhaps the problem asks "a condição que cos θ deve satisfazer" singular, but may include n. Could be "cosθ = \frac{L q B}{2\pi n m v}, n\in\mathbb{N}^*". That seems mathematically correct. Let's derive with n included, and maybe mention n = 1,2,3,... in answer. Let's ensure part c "em função dos dados do problema" not "em função de n"? The data include L, q, B, m, V maybe. n is not data but a positive integer. So answer should include n as a natural number. Maybe acceptable. Let's also consider the trajectory in the magnetic field: Because initial velocity has component parallel, the particle moves in a helix; it doesn't return to the same field line after a full period? Wait, the projection in transverse plane is a circle centered at guiding center. The initial point is on the circle. After one period, it returns to same transverse position. Yes. But if there is no electric field, the guiding center is fixed? Actually for a charged particle in uniform B, the guiding center is \( \mathbf{R} = \mathbf{r} + \frac{\mathbf{v}\times \mathbf{B}}{\omega B} \)? Let's compute: If B along x, v_perp = (0, v_y, v_z). The orbit is a circle; after one period, transverse position returns to initial. So yes. Let's compute radius maybe not needed. Let's formulate answer in Portuguese. Potential final response: a) Pelo teorema da energia cinética: \(qV = \frac12 m v^2\), logo \(v = \sqrt{\frac{2qV}{m}}\). b) A força magnética é \( \vec F = q \vec v \times \vec B\). Ela é perpendicular à velocidade, portanto não altera o módulo da velocidade; a componente \(v_\parallel = v\cos\theta\) permanece constante ao longo de \(\vec B\), enquanto a componente perpendicular \(v_\perp = v\sin\theta\) descreve movimento circular uniforme de raio \(R = \frac{m v \sin\theta}{qB}\). Assim, a trajetória é uma hélice cilíndrica (movimento helicoidal uniforme), resultante da composição de um MRU na direção do campo com um MCU no plano perpendicular. c) O período do movimento circular é \(T = \frac{2\pi m}{qB}\). Em cada período, a partícula avança ao longo do campo uma distância \(p = v_\parallel T = v\cos\theta \frac{2\pi m}{qB}\). Para que ela retorne à mesma linha de campo no ponto 2, situado a uma distância L do ponto 1, deve-se ter \(L = n p\), onde \(n=1,2,3,\ldots\). Portanto: \[ \cos\theta = \frac{L q B}{2\pi n m v} = \frac{L q B}{2\pi n \sqrt{2mqV}}? \] Wait check: v = sqrt(2qV/m), so m v = sqrt(2 m q V). Yes denominator \(2\pi n \sqrt{2 m q V}\). But let's verify: \(m v = m \sqrt{2qV/m} = \sqrt{2m^2 qV/m}? = \sqrt{2 m q V}\). Yes. So \(\cos\theta = \frac{L q B}{2\pi n \sqrt{2 m q V}}\). Alternatively \( \frac{L B}{2\pi n}\sqrt{\frac{q}{2mV}}\). Which is more elegant? Let's pick one. Let's check dimensions: qB/(m v) = (C T)/(kg m/s) =? T = kg/(C s), so qB = C*kg/(C s)=kg/s; denominator m v = kg m/s; ratio=(kg/s)/(kg m/s)=1/m? Wait cos dimensionless; L q B/(m v) = m*(kg/s)/(kg m/s)=? kg/s / (kg m/s)=1/m. Then L times 1/m dimensionless. Good. qB/(m v) = (C * kg/(C s))/(kg m/s)= (kg/s)/(kg m/s)=1/m. Yes. Alternative q/(sqrt(2 m q V)) dimension? q C; sqrt(kg C V)= sqrt(kg C * J/C?) J=kg m^2/s^2; V=J/C=kg m^2/(C s^2); kg C V = kg C * kg m^2/(C s^2)=kg^2 m^2/s^2 => sqrt = kg m/s = momentum. So q / momentum = C/(kg m/s); times L B? B T=kg/(C s); L B q/momentum = m*(kg/(C s))*C/(kg m/s)=? kg/s / (kg m/s)=1/m times m? Actually m* (kg/(C s))*C/(kg m/s) = m*(kg/s)/(kg m/s)=1. Good. Let's include n in answer. Maybe if they expect "cosθ = L q B/(2π m v)" without n, would lose? Let's see: The problem says "Em um desses retornos, ela passa pelo ponto 2"; if point 2 is at distance L and it's "um desses retornos", then it could be first return? But "um desses" means "one of these"; however, in physics problems, they may not want n because point 2 could be after the first return? But if L is arbitrary, they may ask condition that cosθ must satisfy "para que a partícula passe pelo ponto 2" not "pela primeira vez". Since n can be any positive integer, including 1. If they wanted first return, they'd say "no primeiro retorno". So include n. Let's consider if point 2 is located at distance L from point 1 along the same field line, but the particle may pass through point 2 at a return that is not the first. The condition is that there exists n such that L = n*pitch. If question asks "a condição que cosθ deve satisfazer", maybe answer is \(\cos\theta = \frac{L q B}{2\pi m v}\) with the understanding that \(L\) is the distance corresponding to one return? But "em um desses retornos" definitely means some return, not necessarily first. Let's search memory: ITA 2024 Física? I think the official answer might be \( \cos\theta = \frac{L q B}{2\pi m v} \sqrt{\frac{m}{2qV}}\)? Hmm. Let's parse "a partícula volta a passar pela mesma linha de campo na qual o movimento teve início. Em um desses retornos, ela passa pelo ponto 2, situado a uma distância L do ponto 1." If it "volta a passar" in singular? Actually "volta a passar pela mesma linha de campo" could be a recurring event. "Em um desses retornos" = in one of these returns. Therefore they likely intend n. However, in many textbook solutions, they define "p" pitch and say "para que passe pelo ponto 2, L = n p, n=1,2,3,...". Then cosθ condition includes n. But maybe they ask "a condição que cosθ deve satisfazer" and answer "cosθ = L q B/(2π n m v), n=1,2,...". Let's do that. Let's also consider if there is another possibility: The particle could pass through point 2 after an integer number of turns plus a partial turn? No, because point 2 is on the original field line, transverse displacement zero; that only occurs at full periods (unless velocity perpendicular component zero? But θ not 0). So integer n. Let's ensure not missing factor 1/2? Period T = 2π m/(qB), yes. In one period, time T = 2π/ω = 2π m/(qB). Distance along B = v cosθ * T. Good. Let's craft final answer with development item by item. Need final line format: "Resposta final: a) ...; b) ...; c) ...". Short. Let's write in Portuguese. Potential final: a) \(qV = \frac12 m v^2 \Rightarrow v = \sqrt{\frac{2qV}{m}}\). b) Movimento helicoidal uniforme (hélice cilíndrica). Justificativa: a força magnética é perpendicular à velocidade e não realiza trabalho; a componente \(v_\parallel = v\cos\theta\) não sofre força e permanece constante, enquanto a componente \(v_\perp = v\sin\theta\) realiza MCU num plano perpendicular a \(\vec B\). Logo, a partícula descreve uma hélice de raio \(R = \frac{m v\sin\theta}{qB}\). c) O período do MCU é \(T = \frac{2\pi m}{qB}\). Em cada volta, o deslocamento paralelo ao campo é \(p = v\cos\theta T = \frac{2\pi m v\cos\theta}{qB}\). Para que passe pelo ponto 2, a distância L deve ser um múltiplo inteiro de p: \[ L = n p \Rightarrow \cos\theta = \frac{L q B}{2\pi n m v} = \frac{L q B}{2\pi n\sqrt{2mqV}}, \quad n=1,2,3,\ldots \] Wait simplification: \(m v = m sqrt(2qV/m) = \sqrt{2m q V}\). Yes. Let's verify simplification: \(m v = \sqrt{2m^2 q V/m}?\) Let's do carefully: v = sqrt(2qV/m). m v = m * sqrt(2qV/m) = sqrt(m^2 * 2qV/m) = sqrt(2m q V). Yes. Then denominator 2π n m v = 2π n sqrt(2 m q V). So cosθ = L q B / (2π n sqrt(2 m q V)). Good. Could also write \( \cos\theta = \frac{L B}{2\pi n}\sqrt{\frac{q}{2mV}}\). But first is fine. Let's make sure no error: \(m v\) multiplied by qB: qB * L / (2π n * sqrt(2 m q V)). Dimension mass? sqrt(2 m q V) has dimensions sqrt(kg C J/C)=sqrt(kg J)=sqrt(kg * kg m^2/s^2)=kg m/s = momentum. qB L? qB dimension C * kg/(C s)=kg/s; times L m = kg m/s; divide momentum = dimensionless. Good. Let's final. But we should include "n" in final line? Maybe "c) cosθ = ... , n∈N*". Yes. Let's also address part b with "movimento helicoidal uniforme" not "helicoidal". In Portuguese "movimento helicoidal uniforme" or "hélice cilíndrica". Good. Let's produce final answer with appropriate formatting.

**Analyzing Physics Problem** I am currently focused on understanding the physics problem presented, specifically the text within Image 1 and how it relates to the ITA 2027 exam context. My next steps involve deciphering any provided equations and information that will be crucial for solving the problem. **Deconstructing Experimental Setup** I'm working through the specifics of the mass spectrometer experiment. My current focus is on interpreting the given parameters for particle acceleration and their entry into the magnetic field region. The angle of entry relative to the magnetic field is a key consideration. **Calculating Particle Velocity** I’m calculating the particle’s speed as it enters the magnetic field, using energy conservation based on the accelerating potential difference. My next focus is on characterizing its motion within the magnetic field. **Analyzing Helical Motion** I'm now synthesizing the components of motion. The particle exhibits uniform motion along the magnetic field and uniform circular motion perpendicular to it, resulting in a helical trajectory. My next step is to apply this understanding to the specific problem parameters. **Determining Trajectory Condition** I've identified the motion as uniform helical, and I'm now focused on item c) of the problem. My current thinking centers on the condition required for the particle to return to the same magnetic field line, specifically considering the radius of circular motion in relation to the period of revolution. **Clarifying Helix Pitch** My current focus is on clarifying that point 1 refers to a line on the cylinder's surface, not an axis. I’ve confirmed the particle intersects the original field line at discrete times $t = nT$. My next step is to verify if the helix intersects the same field line within a single period. **Deriving Angle Condition** I am now deriving the specific condition for $\cos\theta$ required for the particle to reach point 2. My current work involves substituting the expression for velocity and ensuring the resulting equation satisfies the physical constraints on the cosine of an angle. **Investigating Angle Constraints** I am now reconciling the derived angle condition with the provided small-angle approximation. My thoughts are focused on understanding why this approximation was offered and if it directly applies to this specific problem's constraints, prompting a review of the problem's context and origin. **Confirming Exam Year** I am now trying to pinpoint the exact exam year based on stylistic cues and specific phrasing, considering ITA 2023-2026 and IME. My focus is on the context of the small-angle approximation provided. **Revisiting Problem Statement** I'm re-evaluating the initial problem statement regarding the small-angle approximation. My current focus is on whether this approximation is truly necessary or if it was provided as a potential hint for a specific solution path. I will now analyze parts a) and b) of the question in detail. **Refining Angle Calculation** I am now focusing on the specific request for $\cos\theta$ instead of $\theta$ itself. My current derivation shows $\cos\theta = \frac{q B L}{2 \pi n m v}$, which aligns with the structure required by item c), and I am investigating the purpose of the provided small-angle approximation. **Analyzing Angle Constraints** I am now reconciling the derived angle condition with the provided small-angle approximation. My thoughts are focused on understanding why this approximation was offered and if it directly applies to this specific problem's constraints. The condition for $\cos\theta$ is asked for directly, and it appears the small-angle approximation might not be necessary. **Confirming Focusing Principle** I am now confirming that the problem describes magnetic focusing. The condition for particles to return to the same field line relates directly to this principle, and the mention of a mass spectrometer hints at this mechanism. My thoughts are on whether the small-angle approximation is a general characteristic of focusing or specific to this problem. **Re-evaluating Approximation Use** I am now scrutinizing the prompt for item c) more closely. It specifically asks for the condition on $\cos\theta$, implying it's not simply a fixed value like 1 from a small-angle approximation. My current thinking is that the approximation might be a general note on the exam page, not a strict requirement for this specific question, as assuming $\theta \ll 1$ would render the request for a condition on $\cos\theta$ moot. **Analyzing Particle Motion** I am now analyzing the detailed breakdown of items a) and b). For part a), I've confirmed the calculation of particle speed using potential difference and energy conservation. For part b), I am currently defining the coordinate system and decomposing the initial velocity vector to analyze the subsequent motion within the magnetic field. **Analyzing Velocity Components** I am now decomposing the particle's velocity vector into components parallel and perpendicular to the magnetic field. My current focus is on how the Lorentz force influences these components, leading to uniform motion along the field and circular motion in the perpendicular plane. **Analyzing Helix Geometry** I am now examining the geometry of the helical path relative to the magnetic field lines. My focus is on the relationship between the helix pitch, the radius of circular motion, and the angle of velocity to the field, as this is critical for determining when the particle returns to its initial field line. **Deriving Angle Condition** I'm now focusing on the specific request for $\cos\theta$. My derivation yields $\cos\theta = \frac{B L}{2\pi n v} \sqrt{\frac{q}{2mV/m}}$, and I am investigating how the given small-angle approximation might relate to this, if at all. **Confirming Helix Conditions** I am now confirming the conditions for the helical trajectory. My current focus is on the specific constraints on the integer $n$ required for $\cos\theta$ to be physically valid, and reconciling the derived expression with any potential implications of the small-angle approximation provided in the problem. **Verifying Derived Conditions** I am now verifying the derived expression for $\cos\theta$ and its units, ensuring it aligns with the problem's parameters. My focus is on confirming that the small-angle approximation's presence, while perhaps not directly used in the final answer, might relate to the broader context or specific exam conventions for such problems.

**Analyzing Physics Problem** I am carefully reading the physics problem, which appears to be inspired by a mass spectrometer experiment. My immediate focus is on understanding the details presented in the initial description. **Deconstructing Physics Problem** I'm now breaking down the physics problem, which describes charged particles accelerated by a potential difference and then entering a uniform magnetic field. My current task is to identify the key physics principles and information required for each part of the question. **Calculating Particle Speed** I have successfully derived the expression for the particle's speed after acceleration, utilizing the work-energy theorem. The next step involves analyzing its motion within the magnetic field, specifically focusing on the components of its velocity. **Analyzing Particle Motion** I've determined that the motion within the magnetic field consists of both rectilinear and circular components, resulting in a helical trajectory. My current focus is on item (c), specifically the condition for the particle to return to its starting point, which seems to involve the relationship between its velocity and the magnetic field parameters. **Deriving Return Condition** I've established that for the particle to return to the initial field line, its motion must complete an integer number of revolutions perpendicular to the magnetic field. I've derived the condition involving velocity components, magnetic field strength, and the length L.

**Clarifying geometric interpretation** I'm working on deriving based on perpendicular motion and parallel velocity. It’s important to express the final result in a compact form. I need to make sure that I'm considering the geometric interpretation carefully, especially with parameters like ‘c’ possibly being any positive value. I'll keep in mind that cosine of theta must be less than or equal to 1, and that theta should be greater than 0. I’ll wrap this up nicely.