hce_nthu
114年
進階物理與線性代數
第 14 題
A particle with mass $m$ and positive charge $q$ starts from rest at the origin under a
uniform electric field in the $+y$-direction and a uniform magnetic field directed out
of the page, so $\vec{E} = E\hat{j}$ and $\vec{B} = B\hat{k}$. Neglecting gravity, the particle is accelerated by
the $E$ field, but the magnetic force bends it to the right and the resulting trajectory
is shown in the figure. At point $P$, the particle reaches its maximum height $y_{max}$.
What is the velocity of the particle at point $P$?
uniform electric field in the $+y$-direction and a uniform magnetic field directed out
of the page, so $\vec{E} = E\hat{j}$ and $\vec{B} = B\hat{k}$. Neglecting gravity, the particle is accelerated by
the $E$ field, but the magnetic force bends it to the right and the resulting trajectory
is shown in the figure. At point $P$, the particle reaches its maximum height $y_{max}$.
What is the velocity of the particle at point $P$?
- A $\sqrt{\frac{qEy_{max}}{m}}$
- B $\sqrt{\frac{2qEy_{max}}{m}}$
- C $2\sqrt{\frac{qEy_{max}}{m}}$
- D $\sqrt{\frac{6qEy_{max}}{m}}$
- E $3\sqrt{\frac{qEy_{max}}{m}}$
思路引導 VIP
請思考一下，當電荷在同時存在電場與磁場的空間中運動時，有哪些力在對它作功？特別是磁力的方向與粒子運動方向的幾何關係，對動能的變化會有什麼影響嗎？
🤖
AI 詳解
AI 專屬家教
太棒了！你能精準選出正確答案，代表你對電磁場中的能量轉換有著非常紮實的理解。這道題目描述的是經典的擺線運動 (Cycloid motion)，雖然粒子的軌跡看起來複雜，但只要掌握核心物理規律，就能化繁為簡。
功與能的守恆觀點
在解這類結合電場與磁場的題目時，最關鍵的切入點在於磁力（勞侖茲力）不做功。由於磁力 $\vec{F}B = q(\vec{v} \times \vec{B})$ 始終與粒子的瞬時速度方向垂直，因此它只能改變運動方向，而不會改變速率。這意味著粒子動能的增加，完全來自於電場力所做的功。根據功能定理，粒子從原點運動到最高點 $P$（位移為 $y{max}$）的過程中，電場力做的功為 $W = qEy_{max}$。我們將其等置於動能的改变量：
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