Question

A conducting sphere of radius 10cm has unknown charge. If the electric field 20 cm from the centre of the sphere is $1.5 \times {10^3}N/C$ and points radially inward, what is the net charge on the sphere?

Answer 1

Hint: We are given a conducting sphere of certain radius and we have the electric field due to this sphere at a distance which is outside the sphere. We know the formula for calculating electric fields outside a conducting sphere which can be used to calculate the net charge on the conducting sphere.

Formula used:
The electric field due to a conducting sphere can be calculated by the following formula.
$E = \dfrac{1}{{4\pi { \in _0}}}\dfrac{q}{{{r^2}}}{\rm{ }}...\left( i \right)$
Here q is used to represent the net charge on the conducting sphere; r is used to represent the distance from the centre of the sphere at which we are measuring the electric field.
Also we have the constant used in the above equation which is given as
$\dfrac{1}{{4\pi { \in _0}}} = 9 \times {10^9}N{m^2}{C^{ - 2}}$

Detailed step by step solution:
We are given a conducting sphere whose radius is given as
$R = 10cm = 0.1m$
The electric field due to this conducting sphere is measured at a distance of
$r = 20cm = 0.2m$
Therefore, we are calculating electricity outside the sphere.
The magnitude of the electric field measured at distance r is equal to
$E = 1.5 \times {10^3}N/C$
Now we just need to calculate the net charge on the conducting sphere which can be calculated using the expression for the electric field given in equation (i). This can be done in the following way.
$\begin{array}{l}
1.5 \times {10^3} = 9 \times {10^9} \times \dfrac{q}{{{{\left( {0.2} \right)}^2}}}\\
q = \dfrac{{1.5 \times {{10}^3}}}{{9 \times {{10}^9}}} \times {\left( {0.2} \right)^2}\\
q = 6.67 \times {10^{ - 9}}C = 6.67nC
\end{array}$
This is the required amount of net charge on the conducting sphere. This is the required answer to the question.

Note: The formula that we have used above is applicable only when the distance of observation is greater than or equal to the radius of the conducting sphere. Since the conducting sphere is basically a conductor, the electric field inside the conductor is zero.