Torque, Dipole Moment & Potential Energy

PHYMagnetic Dipole & Instruments

A current loop (or magnetic dipole) of moment $\vec{m}=NI\vec{A}$ placed in a uniform field experiences a torque $\vec{\tau}=\vec{m}\times\vec{B}$ that tries to align it with the field, while its orientation energy is $U=-\vec{m}\cdot\vec{B}$ . ISC numericals ask for the dipole moment, the (maximum) torque, and the work done in rotating the dipole between two orientations.

The recurring trap is the angle: $\theta$ in these formulas is always measured between $\vec{m}$ (the normal to the coil) and $\vec{B}$ , not between the plane of the coil and $\vec{B}$ .

Magnetic dipole moment of a coil

m = N I A

N

turns,

I

current,

A

area of the coil; direction along the coil normal by the right-hand rule, unit

\mathrm{A\,m^2}

Torque on a dipole

\tau = m B \sin\theta = N I A B \sin\theta

\theta

angle between

\vec{m}

(coil normal) and

\vec{B}

; maximum torque

mB

\theta=90^{\circ}

, zero at

\theta=0^{\circ}

Potential energy of a dipole

U = -m B \cos\theta

U=-mB

(minimum, stable) at

\theta=0^{\circ}

U=0

\theta=90^{\circ}

U=+mB

(maximum, unstable) at

\theta=180^{\circ}

Work to rotate the dipole

W = mB\,(\cos\theta_1 - \cos\theta_2)

W = U_2 - U_1

, the work done against the field rotating from

\theta_1

\theta_2

; e.g.

0^{\circ}\to90^{\circ}

gives

W=mB

The dipole moment vector $\vec{m}$ points along the coil normal (right-hand rule: curl the fingers along the current, the thumb gives $\vec{m}$ ).
Angle $\theta$ is always between $\vec{m}$ and $\vec{B}$ . If a question gives the angle $\phi$ between the coil plane and $\vec{B}$ , then $\theta = 90^{\circ}-\phi$ .
Torque is maximum when the coil plane is parallel to $\vec{B}$ ( $\vec{m}\perp\vec{B}$ ) and zero when the plane is perpendicular to $\vec{B}$ ( $\vec{m}\parallel\vec{B}$ ).
Stable equilibrium is at $\theta=0^{\circ}$ (minimum energy); unstable equilibrium at $\theta=180^{\circ}$ (maximum energy).
Work to rotate equals the change in potential energy, $W=U_2-U_1=mB(\cos\theta_1-\cos\theta_2)$ ; rotating from alignment to $90^{\circ}$ stores energy $mB$ .
Area must be in $\mathrm{m^2}$ : a $5\ \mathrm{cm}\times4\ \mathrm{cm}$ coil has $A = 20\times10^{-4}\ \mathrm{m^2} = 2\times10^{-3}\ \mathrm{m^2}$ .
In a uniform field the net force on the dipole is zero — only a torque acts; a net force needs a non-uniform field.

Where the marks go

Using the angle between the coil plane and $\vec{B}$ as $\theta$ instead of the angle between the normal $\vec{m}$ and $\vec{B}$ — these differ by $90^{\circ}$ .
Forgetting the number of turns $N$ in $m=NIA$ and hence in the torque $\tau=NIAB\sin\theta$ .
Sign slips in $U=-mB\cos\theta$ — omitting the minus sign makes the stable orientation look like maximum energy.
Not converting the area from $\mathrm{cm^2}$ to $\mathrm{m^2}$ (factor $10^{-4}$ ).

How the board asks it

Numerical $\vec{m}=NI\vec{A}$ and $\tau=NIAB\sin\theta$
A circular coil of $50$ turns and radius $4\ \mathrm{cm}$ carries a current of $2\ \mathrm{A}$ . It is placed in a uniform magnetic field of $0.5\ \mathrm{T}$ such that the plane of the coil makes an angle of $30^\circ$ with the field. Calculate the magnetic dipole moment of the coil and the torque acting on it.
Numericalwork to rotate a dipole, $W=mB(\cos\theta_1-\cos\theta_2)$
A rectangular coil of $100$ turns and area $5\ \mathrm{cm}\times4\ \mathrm{cm}$ carries a current of $0.5\ \mathrm{A}$ in a uniform field of $0.2\ \mathrm{T}$ . Calculate the work done in rotating the coil from the orientation where $\vec{m}$ is aligned with $\vec{B}$ to the orientation where $\vec{m}$ is perpendicular to $\vec{B}$ .
Derive / provetorque on a current loop, $\vec{\tau}=\vec{m}\times\vec{B}$
Derive an expression for the torque experienced by a rectangular current-carrying loop of $N$ turns placed in a uniform magnetic field $\vec{B}$ , and hence express it in terms of the magnetic dipole moment $\vec{m}$ .
Define / state $U=-\vec{m}\cdot\vec{B}$ and net force in a uniform field
State the expression for the potential energy of a magnetic dipole of moment $\vec{m}$ in a uniform field $\vec{B}$ , and explain why the net force on the dipole is zero while a torque still acts on it.
Give reasonsangle convention: $\theta$ between $\vec{m}$ and $\vec{B}$ , not the plane
A student writes the torque on a coil as $\tau=NIAB\sin\phi$ , where $\phi$ is the angle between the plane of the coil and the field. Explain why this is wrong, and state the correct expression in terms of the angle $\theta$ between $\vec{m}$ and $\vec{B}$ .
Assertion–Reason $\tau_{\max}$ when the coil plane is parallel to $\vec{B}$
Assertion: The torque on a current-carrying coil is maximum when the plane of the coil is parallel to the magnetic field. Reason: The torque is maximum when the angle between $\vec{m}$ and $\vec{B}$ is $90^\circ$ . Choose the correct option regarding these two statements.

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.