Answer to Question #232190 in Physics for Soriano

Question

A sphere carrying 5.5 NC of charge is placed 100mm from a small sphere carrying 4.0 NC of charge. Assume the spheres are small relative too the 100mm seperation.

At what position from the particles will the electric magnetic field magnitude be zero

Answer 1

Let $q_1 = 5.5\times 10^{-9}C, \space q_2 = 4\times 10^{-9}C, \space r = 100mm = 0.1m$. The electric field produced by the first sphere at some distance $x$ from it is given as follows:

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

where $k$ is the constant and $x$ is the distance from the center of the first sphere. The second sphere produces the following field at the same point $x$:

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

These fields are directed in opposite direction, thus, the resulting field is zero if

$$E_1 = E_2$$

Solving for $x$, obtain:

$$k\dfrac{q_1}{x^2} = k\dfrac{q_2}{(r-x)^2}\\ q_1(r - x)^2 = q_2x^2\\ q_1r^2 - 2q_1rx + q_1x^2 = q_2x^2\\ (q_1 - q_2)x^2 - 2q_1rx + q_1r^2 = 0\\ 1.5\times 10^{-9}x^2 - 1.1\times 10^{-9}x + 0.055\times 10^{-9} = 0\\ 1.5x^2 - 1.1x + 0.055 = 0\\ x_1 \approx 0.054m = 5.4cm\\ x_2 \approx 0.68m = 68cm$$

Answer. The electric magnetic field magnitude is zero at 5.4cm and 68 cm from the first particle.