In the formula `X = 3YZ^2`, X and Z have dimensions of capacitance and magnetic induction respectively. The dimensions of Y in MKSQ system should be

Options

A. `[M^(-3)L^(-1)T^(3)Q^(4)]`
B. `[M^(-3)L^(-2)T^(4)Q^(4)]`
C. `[M^(-2)L^(-2)T^(4)Q^(4)]`
D. `[M^(-3)L^(-2)T^(4)Q]`

Answer

B

Explanation

Concept of capacity is Q = CV Now by principle of homogeneity.
`[C] = [Q/V] = [Q/(w//Q)] = [Q^(2)/W] = Q^(2)/(ML^(2)T^(-2)) = M^(-1)L^(-2)T^(2)Q^(2)`..... (i)
Here C is capacitance, Q is charge, V is potential and W is work.
Concept of force on current carrying conductor in uniform magnetic field, F = ilB
Now by principle of homogeneity,
`[B] = [F/(il)] = (MLT^(-2))/(QT^(-1) xx L) = MT^(-1)Q^(-1)`........ (ii)
Here B is magnetic induction, F is force, i is electric current and I is length.
`[Y] = [X/Z^(2)] [C/B^(2)]`
By putting dimensions from equations (i) and (ii)
`[Y] = [(M^(-1)L^(-2)Q^(2)T^(2))/(M^(2)Q^(-2)T^(-2))]`
` = [M^(-3)L^(-2)T^(4)Q^(4)]`