Given a charge q, current I and permeability of vacuum $$\mu_0$$. Which of the following quantity has the dimension of momentum ?

Solution :

Dimensions of momentum :

$$[p] = MLT^{-1}$$

Given :

Charge,

$$[q] = AT$$

Current,

$$[I] = A$$

Permeability of vacuum :

From force per unit length between two current carrying wires,

$$\frac{\mu_0 I^2}{r}$$

has dimensions of force.

Therefore,

$$[\mu_0] = \frac{[F][r]}{[I]^2}$$

$$= \frac{(MLT^{-2})(L)}{A^2}$$

$$= MLT^{-2}A^{-2}$$

Now consider quantity :

$$\mu_0 qI$$

Its dimensions are :

$$[\mu_0 qI] = (MLT^{-2}A^{-2})(AT)(A)$$

$$= MLT^{-1}$$

which is the dimension of momentum.

Final Answer :

$$\mu_0 qI$$