46E.
Question
A coil with magnetic moment \(1.45\;{\rm{A}} \cdot {{\rm{m}}^{\rm{2}}}\) is oriented initially with its magnetic moment anti-parallel to a uniform \(0.835\;{\rm{T}}\) magnetic field. What is the change in potential energy of the coil when it is rotated \(180^\circ \) so that its magnetic moment is parallel to the field?
Step-by-Step Solution
VerifiedThe change in potential energy of the coil is \( - 2.42\;{\rm{J}}\).
Magnetic moment of the coil
\(m = 1.45\;{\rm{A}} \cdot {{\rm{m}}^{\rm{2}}}\)
Strength of the magnetic field
\(B = 0.835\;{\rm{T}}\)
The potential energy of a coil of magnetic moment \(m\) in a magnetic field \(B\) is \(U = - mB{\rm{cos}}\theta \) .....(i)
Here, \(\theta \) is the angle between the directions of magnetic moment and magnetic field.
Initially, the magnetic moment was anti-parallel to the magnetic field, that is the angle between them was
\(\theta = 180^\circ \)
From equation (i), the initial potential energy was
\(\begin{aligned}{U_1} = - mB{\rm{cos180}}^\circ \\ = mB\end{aligned}\)
After rotation the magnetic moment is parallel to the magnetic field, that is the angle between them is
\(\theta = 0^\circ \)
From equation (i), the final potential energy was
\(\begin{aligned}{U_2} = - mB{\rm{cos0}}^\circ \\ = - mB\end{aligned}\)
Thus, the change in potential energy is
\(\begin{aligned}{U_2} - {U_1} = - mB - mB\\ = - 2mB\end{aligned}\)
Substitute the values to get
\(\begin{aligned}{U_2} - {U_1} = - mB - mB\\ = - 2 \times 1.45\;{\rm{A}} \cdot {{\rm{m}}^{\rm{2}}} \times 0.835\;{\rm{T}}\\ = - 2.42 \cdot \left( {1\;{\rm{A}} \cdot {{\rm{m}}^{\rm{2}}} \cdot {\rm{T}} \times \frac{{1\;{\rm{J}}}}{{1\;{\rm{A}} \cdot {{\rm{m}}^{\rm{2}}} \cdot {\rm{T}}}}} \right)\\ = - 2.42\;{\rm{J}}\end{aligned}\)
Thus, the required energy is \( - 2.42\;{\rm{J}}\).