Question

The Power \( P \) radiated from an accelerated charged particle is given by \( P \propto \left( \frac{q a}{c^n} \right)^m \), where \( q \) is the charge, \( a \) is the acceleration, and \( c \) is the speed of light in vacuum. From dimensional analysis, the value of \( m \) and \( n \) respectively are:

• A. m = 2, n = 2
• B. m = 2, n = 3
• C. m = 3, n = 3
• D. m = 0, n = 1

Hint:

In dimensional analysis, equate the dimensions of both sides of the equation and solve for the unknown exponents. Remember that dimensionless quantities do not contribute to the overall dimensional equation.

Answer 1

The Correct Option is B

Step 1: In dimensional analysis, ensure the dimensions on both sides of the equation are equivalent. For power \(P\), its fundamental dimensions are: \[ [P] = [ML^2T^{-3}] \] (where \(M\) is mass, \(L\) is length, and \(T\) is time).

Step 2: The dimensions of each variable in the equation \(P \propto \left( \frac{q a^m}{c^n} \right)\) are:

• (a) \([q] = [M^0L^0T^0]\) (dimensionless)
• (b) \([a] = [LT^{-2}]\) (acceleration)
• (c) \([m] = [M]\) (mass)
• (d) \([c] = [LT^{-1}]\) (speed of light)
• (e) \([n] = [L^0T^0]\) (dimensionless)

Step 3: Equating dimensions and solving for \(m\) and \(n\) yields \(m = 2\) and \(n = 3\).