Question

Following figure shows a charge q1 of +1.5 µC and a charge q2 of +2.3µC. The first charge is at the origin of an x-axis, and the second is at a position x = L, where L = 13 cm. At what point P along the x-axis is the electric field zero?

Answer 1

The electric field from the first charge is:

$$E_1 = k\dfrac{q_1}{r^2}$$

where $r$ is the distance from the first charge to the required point.

The electric field from the second charge is:

$$E_2 = k\dfrac{q_2}{(L-r)^2}$$

The total field is zero at the point (minus because they are directed in different sides):

$$E_1 - E_2 = 0\\ k\dfrac{q_1}{r^2} - k\dfrac{q_2}{(L-r)^2} = 0$$

Solving for $r$, obtain:

$$(q_1 - q_2)r^2 - 2q_1Lr + q_1L^2 = 0$$

Substituting numbers and solving the quadratic equation, obtain:

$$-0.8\times 10^{-6}r^2 - 3.9\times 10^{-7}r+ 2.535\times 10^{-8}= 0\\ -80r^2 - 39r + 2.535= 0\\ r \approx 0.058m = 5.8cm$$

Answer. 5.8 cm.