Question

The dielectric strength of air is $3.0 \times {10^6}N{C^{ - 1}}$ .The largest charge that a .3cm radius metal sphere can hold without sparking is :
A. 9nC
B. 8.2nC
C. 6nC
D. 3nC

Answer 1

Hint:For a pure electrically insulating material, the maximum electric field that the material can withstand under ideal conditions without undergoing electrical breakdown and becoming electrically conductive (i.e. without failure of its insulating properties ).

Formulae Used:
$E = \dfrac{Q}{{4\pi {\varepsilon _0}{R^2}}}$ , where Q is the charge in coulomb ,R is the radius of sphere, $\pi $ is pi which has value 3.143 and ${\varepsilon _0}$ is the permittivity of free space having value $8.85 \times {10^{ - 12}}$ Farad/m.
Charge: electric charge is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two type of charges:
Positive and negative charge.

Complete step-by-step solution:
We are being given the value of dielectric strength, radius of the sphere and we have to calculate the charge.
Therefore, using the above formula of hint, we shall calculate charge:
$
   \Rightarrow E = \dfrac{Q}{{4\pi \times 8.85 \times {{10}^{ - 12}} \times {{.3}^2} \times {{10}^{ - 4}}}} \\
   \Rightarrow 3 \times {10^6} = \dfrac{Q}{{11.309 \times 8.85 \times {{10}^{ - 16}}}} \\
   \Rightarrow 30.027 \times {10^{ - 10}} = Q \\
 $ (to convert centimetre square into metre square we multiply centimetre square by ${10^{ - 4}}$ as $1m = 100cm$ )
In the options the value is given in the nano coulomb
Therefore, $Q = 3.02743 \times {10^{ - 9}}$
The value is 3nC.

Note:-
Various factors which effect dielectric strength :
->It decreases with increased sample thickness
->It decreases with increased operating temperature
->It decreases with increased frequency.
->For gases it increases with pressure.
->For air ,dielectric strength increases slightly as the absolute humidity increases but decreases with an increase in relative humidity.
->For gases like nitrogen, sulphur hexafluoride it normally decreases with increased humidity as ions in water can provide conductive channels.