Q. 25.0( 1 Vote )

# Consider three quantities x = E/B, y = and . Here, ℓ is the length of a wire, C is a capacitance and R is a resistance. All other symbols have standard meanings.A. x, y have the same dimension.B. y, z have the same dimensionsC. z, x have the same dimensionsD. None of the three pairs have the same dimensions.

(A)Dimension for electric field E is: F/q = [MLT-3A-1]
Force due to magnetic field is given as: F=qvB.
Where q is the charge, v is the velocity of charge and B is the magnetic field.
Hence The dimensions of charge, q is [AT]

The dimension of the velocity, V is [LT-1]

Putting the value in the above formula, we get

Dimensions of B = Dimensions of B = [MT-2A-1]
Thus, (B) We know that  Where

μ0 is permeability of free space,

ϵ0 is the permittivity of free space and

c is the speed of light.
Thus, y = c, c is the speed and unit of speed is m/s
Dimensions of y = [LT-1]
(C)

Reactance, Where

C is the capacitance

R is the resistance

l is the length of the wire.

In a CR circuit the product CR is the time constant and unit is seconds.
Thus, Dimensions of z = l/CR = m/s= [LT-1]
Hence x, y and z have same dimensions.
Thus, options (A),(B) and (C) are the correct options.

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