Question

A rod with uniformly distributed charge $\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{8}}\mathbf{C}$ is $\mathbf{50}\mathbf{cm}$ long. We need to calculate $\mathit{E}$ at a distance of $\mathbf{1}\mathbf{cm}$ from the midpoint of the rod. Which equation for the electric field of a rod should we use? (1) Exact, (2) Approximate, (3) Either exact or approximate, (4) Neither—we have to do it numerically, (5) Neither—we need to integrate.

Short answer

The approximate equation can be used.

Step-by-step solution

Identification of given data

The charge on the rod is $q=2\times {10}^{-8}\text{C}$

The length of the rod is $L=50\text{cm}$

The distance of the measurement point from the rod is $r=1cm$

Approximation of electric field due to a charged rod

The approximated form of the electric field due to a charged rod on its mid plane is used when the distance of the point where the field is to be obtained from the rod is negligible compared to the length of the rod.

Determination of the method to obtain the field

Since the length of the rod $\left(L=50\text{cm}\right)$ is much greater than the distance of the measurement point $\left(r=1\text{cm}\right)$ , that is

$L\gg r$

the approximate form of the electric field formula can be safely used.

Hence, the approximate equation can be used.