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:
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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.
Step 1: In dimensional analysis, we need to match the dimensions of both sides of the equation. For the power \(P\), we consider its fundamental dimensions: \[ [P] = [ML^2T^{-3}] \] (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: By equating the dimensions of the two sides and solving for \(m\) and \(n\), we find that \(m = 2\) and \(n = 3\).
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