Figure 22-26 shows two charged particles fixed in place on an axis. (a) Where on the axis (other than at an infinite distance) is there a point at which their net electric field is zero: between the charges, to their left, or to their right? (b) Is there a point of zero net electric field anywhere offthe axis (other than at an infinite distance)?

Considering the concept that the electric field lines move toward the negative charge and lines radially move away from the positive charge, the net electric field can be calculated. If the right direction is said to be positive, the net field at point P can be calculated.

Formula:

The electric field due to a charge at a point, $E=\frac{kq}{r^2}$ (i)

For the condition of having a point at which the net field is zero, the distance at which the net field is zero can be calculated using equation (i) as follows:

$\begin{array}{c}{E}_{+q}+E_{-3q}=0\\ \frac{k(+q)}{{d_1}^2}+\frac{k(-3q)}{{d_2}^2}=0\\ \frac{1}{d_1^2}=\frac{3}{d_2^2}\\ d_2=3d_1\end{array}$

Now, consider that the point is at the right hand side of both the charges, and then the point will be at a distance$d_2$from the charge$+q$which will not be three times of$d_1$from charge $-3q$. Thus, there cannot be any point right to both the charges.

Now, considering the point at the left of both the charges will contradict the above calculation case as field line from positive charge will be radially outward from the charge towards the point that implies the net field will subtract each other.

Hence, the point is between the charges on the axis.

As the charges are of different magnitudes, their net electric fields at any other axis will be due to the contribution of their x and y-components respectively.

Hence, there will be no off-axis points where the net field value will be zero.