Question

State the conditions when magnetic potential energy of a magnetic dipole (current carrying coil) kept in uniform magnetic field be minimum and maximum.​

Answer 1

Explanation:

U=−

m

.

B

orU=mBcosθ

U− Magnetic potential energy

m− magnetic moment of the magnetic dipole

B− Uniform magnetic field

Answer 2

Conditions when magnetic potential energy of a magnetic dipole (current carrying coil) kept in uniform magnetic field is minimum and maximum:

The work done to rotate the magnetic dipole in the external field will be stored as the Potential Energy.

$\therefore \: U = \displaystyle \int dW$

$\implies \: U = \displaystyle \int \tau \: d \theta$

$\implies \: U = \displaystyle \int MB \sin( \theta) \: d \theta$

Putting limits:

$\implies \: U = \displaystyle \int_{ {90}^{ \circ} }^{ \theta} MB \sin( \theta) \: d \theta$

$\implies \: U = \displaystyle MB \: \int_{ {90}^{ \circ} }^{ \theta} \sin( \theta) \: d \theta$

$\implies \: U = \displaystyle MB \: \bigg \{- \cos( \theta) \bigg \} _{ {90}^{ \circ} }^{ \theta}$

$\implies \: U = \displaystyle MB \: \bigg \{ \cos( \theta) - \cos( {90}^{ \circ} ) \bigg \}$

$\implies \: U = \displaystyle MB \cos( \theta)$

So, minimum value of U will be when $\theta$ = 180°.

$\implies \: U_{min} = \displaystyle MB \cos( {180}^{ \circ} )$

$\implies \: U_{min} = \displaystyle - MB$

So, max value of U will be when $\theta$ = 0°.

$\implies \: U_{max} = \displaystyle MB \cos( {0}^{ \circ} )$

$\implies \: U_{max} = \displaystyle MB$

Hope It Helps.