Question

In relativistic mechanics, m = $\dfrac{{{m_o}}}{{\sqrt {\left( {1 - \dfrac{{{v^2}}}{{{c^2}}}} \right)} }}$. What is the equivalent relation in electricity for electric charge?
A) q = ${q_o}$
B) q = $\dfrac{{{q_o}}}{{\sqrt {\left( {1 - \dfrac{{{v^2}}}{{{c^2}}}} \right)} }}$
C) ${q_o} = \dfrac{q}{{\sqrt {\left( {1 - \dfrac{{{v^2}}}{{{c^2}}}} \right)} }}$
D) q = $\dfrac{{{q_o}v}}{c}$

Answer 1

Hint:Mass and charge both follow theory of relativity. Mass of a body depends on the frame of reference but charge doesn’t.

Complete step by step solution:
The mass of an object depends on the frame of reference in which it is measured. When the object moves with a speed comparable to the speed of light, we cannot use the value of its mass at rest. The mass of the object is governed by the equation,
$m = \dfrac{{{m_o}}}{{\sqrt {\left( {1 - \dfrac{{{v^2}}}{{{c^2}}}} \right)} }}$
When the speed of the object is very small as compared to the speed of light, the term $\dfrac{v}{c}$ << 1, and its square is an even smaller number. So the denominator is approximately equal to 1, and hence, the mass is taken the same as its mass at rest which is equal to ${m_o}$.
Charge also follows the theory of relativity. However, charge does not depend on the frame of reference from which the object is observed. If the object with a charge ${q_o}$ moves at a speed comparable to the speed of light, the charge on the object does not change because of its independence from the frame of reference. Hence, the charge of the body does not depend upon the speed at which the object is moving. Therefore, the charge remains the same in relativistic mechanics.

The correct answer is option A.

Note:Relativistic mechanics is the branch of mechanics compatible with special relativity and general relativity. It is applicable in all cases but in cases where the velocity of the object is much lesser than the speed of light, it gives the same result as classical mechanics. So, it is used in cases where the speed of the object is comparable to the speed of light.