Question

The dimensional formula of JωL is ______. Take Q as the dimension of charge.

विकल्प

• M¹L²T¹Q^–2

• M¹L²T^–1Q^–2

• M¹L^–2T^–1Q^–2

• M^–1L²T^–1Q^–2

Answer 1

The dimensional formula of JωL is M¹L²T^–1Q^–2. Take Q as the dimension of charge.

Explanation:

ωL being inductive reactance, can be dimensionally equated to resistance,

∴ [R] = [ωL] = [M¹L²T³I^–2]

Now, charge = current × time

∴ [Q] = [IT]

∴ [R] = [M¹L²T^–1Q^–2]  ......(Substituting for I and T in terms of Q)

J is mechanical equivalent of heat and is dimensionless.

∴ [JωL] = [ωL] = [R] = [M¹L²T^–1Q^–2]

Alternate Method:

[L] = `([phi])/([I]) = ([B][L^2])/([A^1]) = ([M^1T^-2A^-1][L^2])/([A^1])`

∴ [L] = [M¹T^–2L²A^–2]

[ω] = rad/sec = [T^–1]

J is mechanical equivalent and hence dimension less.

∴ [JωL] = [M¹L²T^–2A^–2] [T^–1]

∴ [JωL] = [M¹L²T^–3A^–2]  ......(i)

Now, [Q] = [I¹ T¹]

Substituting for I and T in terms of Q in equation (i)

∴ [JωL] = [M¹L²T^–1Q^–2]