Question

Question: Figure 24-45 shows a thin rod with a uniform charge density of$\mathbf{2}\mathbf{.}\mathbf{00}\mathit{\mu }\mathit{c}\mathbf{/}\mathit{m}$. Evaluate the electric potential at point Pif$\mathit{d}\mathbf{}\mathbf{=}\mathbf{}\mathit{D}\mathbf{}\mathbf{=}\mathbf{}\mathit{L}\mathbf{/}\mathbf{4}\mathbf{.}\mathbf{00}$. Assume that the potential is zero at infinity.

Short answer

Answer:

The electric potential at point P is$2.18\times {10}^4\text{Volt}$.

Step-by-step solution

The given data

1. Uniform charge density,$\lambda =2\mu c/m$
2. Distance of the point P from the center of the axis in the figure,$d=D=L/4$
3. The potential is zero at infinity.

Understanding the concept of the electric potential

Using the concept of the electric potential at a point on a thin rod, we can get the individual potential due to each charge. Now, using the sum of these values, we can get the desired values of the potentials at the center and the point considering the distance of the point.

Formulae:

The linear charge density of a distribution, $\lambda =\frac{dq}{dx}$ (i)

The electric potential at a point of a thin rod, $dV=\frac{dq}{4\pi {\epsilon }_{0}r}$ (ii)

Step 3: Calculation of the electric potential

The distance of the point P in the given figure,$r=\sqrt{x^2+d^2}$

At point P, the electric potential is given using value of equation (i) in the integration of the equation (ii) as follows:

Hence, the value of the electric potential is 2.18 x 10⁴volt