Electric Potential — Question 10

Question 10

One accelerating voltage produces distinct energy speed and momentum ratios

Two positive ions have $(q_1,m_1)=(e,m)$ and $(q_2,m_2)=(2e,3m)$. Both start from rest and cross the same potential drop D. Speeds are nonrelativistic.

1. Find their kinetic-energy and speed ratios.

2. Find their momentum ratio and the voltage drop needed for ion 2 to have the same speed ion 1 has after drop D.

Question 10 - Solution

One accelerating voltage produces distinct energy speed and momentum ratios

(a) Method and calculation

$$K_i=q_iD\Rightarrow K_2/K_1=2.$$$$v_i=\sqrt{2q_iD/m_i}\Rightarrow v_2/v_1=\sqrt{2/3}.$$

(b) Interpretation and further calculation

$$p_i=\sqrt{2m_iq_iD}\Rightarrow p_2/p_1=\sqrt6.$$$$\frac{2(2e)D_2}{3m}=\frac{2eD}{m}\Rightarrow D_2=\frac32D.$$

Physical check

Voltage fixes energy per charge, while speed also depends on mass and momentum depends on both mass and energy.