Chapter 4: Q37E (page 715)

Two positive point charges q are placed on the x-axis, one at x = a and one at x = -a. (a) Find the magnitude and direction of the electric field at x = 0. (b) Derive an expression for the electric field at points on the x-axis. Use your result to graph the x-component of the electric field as a function of x, for values of x between -4a and +4a.

Short Answer

Expert verified

The magnitude and direction of the $E = 0$

For $| x | < a : E = (\frac{q}{{πε}_{0}}) (\frac{- ax}{{(x^{2} - a^{2})}^{2}}), for | x | > a : E (\frac{q}{2 {πε}_{0}}) (\frac{x^{2} + a^{2}}{{(x^{2} - a^{2})}^{2}})$

Step by step solution

Data and Equation

Given data;

Two positive charge are place at $x = a$ and $x = - a$

Formula;

The electric field due to charge

$E = \frac{Kq}{a^{2}} \hat{i}$ ........... (1)

Step 2: Find the electric field

(a)

Put $x = a$ in equation (1)

$E_{1} = \frac{Kq}{a^{2}} i$

Put $x = - a$ in equation (1)

$E_{2} = - \frac{Kq}{a^{2}} i$

Total electric filed

$E = E_{1} + E_{2} = \frac{Kq}{a^{2}} i - \frac{Kq}{a^{2}} i = 0$

Hence, the magnitude and direction of the $E = 0$

Find the electric field at

For $x = a$ and $x = - a$ , the electric field

$E_{1} = \frac{q}{4 π ε_{0} (x + a)^{2}} E_{2} = - \frac{q}{4 π ε_{0} (x - a)^{2}}$

Total electric field

$E = E_{1} + E_{2} = \frac{q}{4 {πε}_{0} (x + a)^{2}} - \frac{q}{4 {πε}_{0} (x - a)^{2}} = (\frac{q}{4 {πε}_{0}}) [\frac{(x - a)^{2} + (x + a)^{2}}{((x + a) (x - a))^{2}}] = (\frac{q}{{πε}_{0}}) (\frac{- ax}{{(x^{2} - a^{2})}^{2}})$

Hence, $| x | < a : E = (\frac{q}{π ε_{0}}) (\frac{- a x}{{(x^{2} - a^{2})}^{2}})$

Find the electric field at |x|>a

For $x = a$ and $x = - a$ , the electric field

$E_{1} = \frac{q}{4 π ε_{0} (x - a)^{2}} \begin{aligned} E_{2} & = - \frac{q}{4 π ε_{0} (x + a)^{2}} \end{aligned}$

Total electric field

$\begin{aligned} E & = E_{1} + E_{2} \end{aligned} = \frac{q}{4 π ε_{0} (x - a)^{2}} - \frac{q}{4 π ε_{0} (x + a)^{2}} = (\frac{q}{4 π ε_{0}}) [\frac{(x - a)^{2} + (x + a)^{2}}{((x + a) (x - a))^{2}}] = (\frac{q}{2 π ε_{0}}) (\frac{x^{2} + a^{2}}{{(x^{2} - a^{2})}^{2}})$