Question

A charge of 6.00 pCis spread uniformly throughout the volume of a sphere of radius r = 4.00 cm. What is the magnitude of the electric field at a radial distance of

(a) 6.00 cmand

(b) 3.00 cm?

Short answer

1. The magnitude of the electric field at a radial distance is 15.0 N/C.
2. The magnitude of the electric field at a radial distance is 25.3 N/C.

Step-by-step solution

The given data

1. Charge of the sphere, q = 6.00 pC.
2. The radius of the sphere, R = 4.00 cm.

Understanding the concept of the electric field

First, we compare the radial distances with the radius of the sphere. So, using the formula of the electric field inside the sphere and outside the sphere, we can get the required values at the radial distances.

Formulae:

The magnitude of the electric field at a point r due to charged particles outside the sphere,

$E=\frac{\left|q\right|}{4{\mathrm{\pi \epsilon }}_0{\mathrm{r}}^2}$E= (i)

The magnitude of the electric field at a point r due to charged particle inside the sphere,

$E=\frac{\left|q\right|}{4{\mathrm{\pi \epsilon }}_0{\mathrm{r}}^2}$ (ii)

a) Calculation of the electric field at radial distance r = 6 cm

From the given data, we can see that this point is outside the sphere ( r = 6 cm). Thus, the magnitude of the electric field is given using equation (i) as follows:

$$\begin{gathered} \mathrm{E}=\frac{(9.0\times {10}^9{\mathrm{Nm}}^2/{\mathrm{C}}^2)(6.00\times {10}^{-12}\mathrm{C})}{{(0.060\mathrm{m})}^2} \\ =15.0\mathrm{N}/\mathrm{C} \end{gathered}$$

Hence, the value of the electric field is 15.0 N/C.

b) Calculation of the electric field at radial distance r = 3cm

From the given data, we can see that this point is inside the sphere . Thus, the magnitude of the electric field is given using equation (i) as follows:

$$\begin{gathered} \mathrm{E}=\frac{(9.0\times {10}^9{\mathrm{Nm}}^2/{\mathrm{C}}^2)(6.00\times {10}^{-12}\mathrm{C})(0.030m)}{{(0.060\mathrm{m})}^2} \\ =25.3\mathrm{N}/\mathrm{C} \end{gathered}$$

Hence, the value of the electric field is $25.3\mathrm{N}/\mathrm{C}$.