## Mastering Physics Solutions: Speed of an Electron in an Electric Field

Speed of an Electron in an Electric Field

Part A = 8,577,000 m/s

Solution Below:

Two stationary positive point charges, charge 1 of magnitude 4.00 nC and charge 2 of magnitude 1.95 nC, are separated by a distance of 57.0 cm. An electron is released from rest at the point midway between the two charges, and it moves along the line connecting the two charges.

Part A

What is the speed v(final) of the electron when it is 10.0 cm from charge 1?

This problem is kind of complicated. You can solve it by finding the difference between the potential energy of the electron at the beginning and the end and converting that to kinetic energy (and from there to velocity). Start by finding the potential energy of the electron midway between the two charges (the radius will be 0.57m / 2 = 0.285m):

Vi = kQ1/r + kQ2/r
Vi = k(Q1 + Q2)/r
Vi = k * (4.00nC + 1.95nc)/ 0.285

Convert to coulombs (from nC) and solve:

Vi = 2.08772 * 10^-8 * k
Vi = 2.08772 * 10^-8 * (8.998 * 10^9)
Vi = 187.853 V

Now find the beginning potential energy of the electron:

Ui = q * Vi
Ui = 1.602*10^-19 * 187.853
Ui = 3.0094*10^-17

Now do the same thing, but for the position of the electron that this problem is asking for (10.0 cm from charge 1). The radius will be different for each charge this time around:

Vf = kQ1/r1 + kQ2/r2
Vf = k * 4.00nC / 0.1 + k * 1.95nC / 0.47

Convert to coulombs again and solve:

Vf = 397.252 V

Now find the final potential energy of the electron:

Uf = q * Vf
Uf = 1.602*10^-19 * 397.252
Uf = 6.3640 * 10^-17

Now solve for kinetic energy (use the absolute value of the difference in potential energies):

KE = abs(Ui – Uf)
KE = abs((3.0094*10^-17) – (6.3640 * 10^-17))
KE = 3.3546 * 10^-17

Convert to velocity (the mass of an electron is 9.1094 * 10^-31 kg):

KE = 1/2mv^2
3.3546 * 10^-17 = 1/2mv^2
6.7092 * 10^-17 = mv^2
v^2 = 7.3651 * 10^13
v = 8,577,000 m/s (difference due to rounding)

8,577,000 m/s

### 2 Responses to Mastering Physics Solutions: Speed of an Electron in an Electric Field

1. tegelee says:

I’m not sure why you did (q1+q2)/r first and then multiplied by k last, because order of operations states that you should do q1+q2 first and then do k+ANS/r in that order. so that answer is already wrong, thus making your entire answer and formula plug in at the top wrong. Sorry.

• tegelee says:

k*ANS/r (not +)