## The resultant magnetic moment for the following arrangement is

Q. The resultant magnetic moment for the following arrangement is (a) M

(b) 2M

(c) 3M

(d) 4M

Ans: (b)

Sol: M1 = 2 M ; M2 = 2 M ; θ = 120°

Resultant Magnetic moment

$\displaystyle M_R = \sqrt{M_1^2 + M_2^2 + 2 M_1 M_2 cos\theta}$

$\displaystyle M_R = \sqrt{(2M)^2 + (2M)^2 + 2 \times 2M \times 2M \times cos120}$

$\displaystyle M_R = \sqrt{4M^2 + 4M^2 + 2 \times 2M \times 2M \times (-\frac{1}{2})}$

$\displaystyle M_R = \sqrt{4 M^2}$

MR = 2 M

## A thin bar magnet of length ‘l’ and magnetic moment ‘M’ is bent at the mid point so that the two parts are at right angles….

Q. A thin bar magnet of length ‘l’ and magnetic moment ‘M’ is bent at the mid point so that the two parts are at right angles. The new magnetic length and magnetic moment are respectively

(a) √2l , √2M

(b) l/√2 , M/√2

(c)  √2l , M /√2

(d) l/√2 , √2M

Ans: (b)

Sol: Net magnetic moment

$\displaystyle M_R = \sqrt{(\frac{M}{2})^2 + (\frac{M}{2})^2}$

$\displaystyle M_R = \sqrt{\frac{M^2}{2}} = \frac{M}{\sqrt{2}}$

Effective length $\displaystyle = \sqrt{(\frac{l}{2})^2 + (\frac{l}{2})^2}$

$\displaystyle = \sqrt{\frac{l^2}{2}} = \frac{l}{\sqrt{2}}$

## A bar magnet of magnetic moment M is bent in ‘⊔’ shape such that all the parts are of equal lengths. Then new…..

Q. A bar magnet of magnetic moment M is bent in ‘⊔’ shape such that all the parts are of equal lengths. Then new magnetic moment is

(a) M/3

(b) 2M

(c) √3 M

(d) 3√3 M

Ans: (a)

Sol: When a thin bar magnet of magnetic moment M is bent in the form of ⊔- shape with the arms of equal length , then

Magnetic moment of Each part = M/3

Net magnetic moment of the combination,

$\displaystyle \vec{M’} = \frac{M}{3}(-\hat{j}) + \frac{M}{3}(\hat{i}) + \frac{M}{3}(\hat{j})$

M’ = M/3