Mastering Physics Solutions: Stopping the Proton

Stopping the Proton

Solution Below:

An infinitely long line of charge has a linear charge density of 8.00*10^−12 C/m. A proton is at distance 19.0 cm from the line and is moving directly toward the line with speed 1200 m/s.

Part A

How close does the proton get to the line of charge?

To solve, just equate kinetic energy with electric potential energy. As the proton gets closer to the field, it will slow down and its kinetic energy will decrease, eventually becoming zero. At this point, where the proton is stationary in front of the field, it will also have zero potential energy. The math behind this requires some calculus, but a simplified formula you can use is:

r = d * e^(-mv^2 / (4kλq)

Where r is the radius you’re solving for, d is the initial distance, m is the mass of the proton, v is the velocity of the proton, k is coulomb’s constant, λ is the linear charge density and q is the charge of the proton:

In case you need the constants, q = (1.602 * 10^-19), k = (8.9875 * 10^9) and the mass of a proton is 1.6726 * 10^-27 kg

r = d * e^(-mv^2 / (4kλq)

r = 0.19 * e^(-(1.6726 * 10^-27) * (1200)^2 / (4 * (8.9875 * 10^9) * (8.00*10^−12) * (1.602 * 10^-19))

r = 0.19 * e^(2.4085^-21 / 4.6074 * 10^-20)
r = 0.19 * e^(0.052276)
r = 0.1803 m

0.180 m