Capacitors

JEE Main level, single correct, medium

122 of 180

Question 122 of 180

The dimensional formula of 12ϵ0E2\dfrac{1}{2}\epsilon_0 E^2 (ϵ0\epsilon_0 = permittivity of vacuum and EE = electric field) is MaLbTcM^a L^b T^c. The value of 2a−b+c=2a - b + c = _______.

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Answer: (B) 11

The quantity 12ϵ0E2\dfrac{1}{2}\epsilon_0 E^2 represents the energy density of an electric field.

Energy density is defined as energy per unit volume.

Dimensional formula of energy is [ML2T−2][M L^2 T^{-2}].

Dimensional formula of volume is [L3][L^3].

Dimensional formula of energy density =[ML2T−2][L3]=[ML−1T−2]= \dfrac{[M L^2 T^{-2}]}{[L^3]} = [M L^{-1} T^{-2}].

Comparing this with MaLbTcM^a L^b T^c, we get:
a=1a = 1
b=−1b = -1
c=−2c = -2

The value of 2a−b+c=2(1)−(−1)+(−2)=2+1−2=12a - b + c = 2(1) - (-1) + (-2) = 2 + 1 - 2 = 1.

Answer: 11

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