Question

A sphere of radius 1cm has a potential of 8000V. Then the energy density near its surface will be.......(in MKS).
A. 31.87
B. 63.43
C. 8.52
D. 2.83

Answer 1

Hint: In this question, first of all, from the given electric potential of the sphere , we will calculate the electric field near the surface of the sphere. After this we will use the formula to calculate the energy density of the electric field.
Formula used- E = $\dfrac{{\text{V}}}{R}$ and U = $\dfrac{1}{2}{ \in _0}{{\text{E}}^2}$

Complete Step-by-Step solution:
A sphere is given having radius = R =1cm

The electric potential at the surface of sphere = 8000V
We know that the relation between electric field and electric potential is given as:
E =$\dfrac{{\text{V}}}{R}$
Putting the value of ‘V’ and ‘R’ in above equation, we get:
E =$\dfrac{{8000}}{1}$V/cm = 8000V/cm=$8 \times {10^5}$V/m

Therefore the electric field at surface of sphere = 8000V/cm= $8 \times {10^5}$V/m

Now, we know that the formula for calculating the energy density of electric field is given as:
U = $\dfrac{1}{2}{ \in _0}{{\text{E}}^2}$ , where
${ \in _0}$ is permittivity of free space = $8.85 \times {10^{ - 12}}{m^{ - 3}}k{g^{ - 1}}{s^4}{A^2}$
E = electric field at surface of sphere = $8 \times {10^5}$V/m

Putting the values in above equation, we get:
U=\[\dfrac{1}{2} \times 8.85 \times {10^{ - 12}} \times {\left( {8 \times {{10}^5}} \right)^2} = 8.85 \times 32 \times {10^{ - 2}} = 2.832\]$J/{Kgs}$.
So, option D is correct.

Note - Energy Density is defined as the total amount of energy in a system per unit volume. For example, the number of calories per gram of food. Foodstuffs that have low energy density provide less energy per gram of food which means that you can eat more of them since there are fewer calories.
It is denoted by letter U.
Magnetic and electric fields can also store energy.
In the case of the electric field or capacitor, the energy density formula is given by
U = $\dfrac{1}{2}{ \in _0}{{\text{E}}^2}$
E = E is the electric field
ε0 = permittivity of free space
The energy density formula in case of magnetic field or inductor is given by,
${\text{U = }}\dfrac{{{B^2}}}{{2{\mu _0}}}$
B =Magnetic field
μ 0 =permeability of free space
Regarding electromagnetic waves, both magnetic and electric fields are equally involved in contributing to energy density. Therefore, the formula of energy density is the sum of the energy density of the electric and magnetic field.