Question

A proton traveling along the x-axis enters a region at x = 0 where the x-component of the electric field is given by E = Ao/x1/2. How much work is done on the proton by the electric field as it moves along the x-axis from x = 0 to x = 7 m?

Answer 1

.Answer:

The value of the work done is $\bf{ 5.29 qA_{0}}$.

Explanation:

When a charged particle having charge $q$ is moving through an electric field $E$, the net force ($F$) on the charge is

$F = qE~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)$

and the work done ($W$) by the particle is

$W = \int\limits^x_0 {F} \, dx ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2)$

Given, $E = \dfrac{A_{0}}{x^{1/2}}$.

Substitute the value of electric field in equation (1) and then substitute the result in equation (2).

$W &=& \int\limits^7_0 {q\dfrac{A_{0}}{x^{1/2}}} \, dx \\&=& qA_{0} \int\limits^7_0 {x^{-1/2}} \, dx \\&=& 2qA_{0}[x^{1/2}]_{0}^{7}\\&=& 5.29 qA_{0}$