(II) Two point charges, \({{\bf{Q}}_{\bf{1}}}{\bf{ = - 32}}\;{\bf{\mu C}}\) and \({{\bf{Q}}_{\bf{2}}}{\bf{ = + 45}}\;{\bf{\mu C}}\) are separated by a distance of 12 cm. The electric field at point P (see Fig. 16–57) is zero. How far from \({{\bf{Q}}_{\bf{1}}}\) is P?

Whether at rest or in motion in an electric field, a charged particle experience a force due to the electric field. The electric field changes the kinetic energy of the particles.

The magnitude of the first charge is\({Q_1} = - 32\;{\rm{\mu C}}\).

The magnitude of the second charge is \({Q_2} = + 45\;{\rm{\mu C}}\).

The separation between both charges is \(d = 12\;{\rm{cm}}\).

The expression for the electric field due to point charge \({Q_1}\) at point P can be written as follows:

\({E_1} = \frac{{k\left| {{Q_1}} \right|}}{{{x^2}}}\)

Here, \(k\) is the Coulombs constant, and \(x\) is the distance of the first charge from point P.

The expression for the electric field due to point charge \({Q_2}\) at point P can be written as follows:

\({E_2} = \frac{{k\left| {{Q_2}} \right|}}{{{{\left( {d + x} \right)}^2}}}\)

If the net electric field is zero at point P, the electric fields due to first and second charges need to be equal in magnitude and opposite in direction. Therefore, it is written as follows:

\(\begin{aligned}{c}{E_1} = {E_2}\\\frac{{k\left| {{Q_1}} \right|}}{{{x^2}}} = \frac{{k\left| {{Q_2}} \right|}}{{{{\left( {d + x} \right)}^2}}}\\\frac{{\left| {{Q_1}} \right|}}{{{x^2}}} = \frac{{\left| {{Q_2}} \right|}}{{{{\left( {d + x} \right)}^2}}}\end{aligned}\)

Substitute the values in the above equation.

\(\begin{aligned}{c}\frac{{\left| { - 32} \right|}}{{{x^2}}} = \frac{{\left| { + 45} \right|}}{{{{\left( {12 + x} \right)}^2}}}\\32{\left( {12 + x} \right)^2} = 45{x^2}\\32\left( {144 + {x^2} + 24x} \right) = 45{x^2}\\4608 + 32{x^2} + 768x = 45{x^2}\\13{x^2} - 768x - 4608 = 0\end{aligned}\)

On solving the above quadratic equation,

\(x = 64.56\;{\rm{cm}} \approx {\rm{65}}\;{\rm{cm}}\).

Thus, the distance of point P from the charge \({Q_1}\) is 65 cm.