Torque on Current Loop — Revision Notes

⚡ 30-Second Revision

Force on current segment: — $d\vec{F} = I(d\vec{l} \times \vec{B})$
Magnetic Dipole Moment: — $\vec{M} = NIA \hat{n}$ (magnitude $M=NIA$)
Torque on Current Loop: — $\vec{\tau} = \vec{M} \times \vec{B}$
Magnitude of Torque: — $\tau = NIAB \sin\theta$

* $\theta$: Angle between $\vec{M}$ (normal to loop) and $\vec{B}$. * If angle between plane and $\vec{B}$ is $\alpha$, then $\theta = 90^\circ - \alpha$.

Maximum Torque: — $\tau_{max} = NIAB$ (when $\theta = 90^\circ$, plane parallel to $\vec{B}$)
Zero Torque: — $\tau = 0$ (when $\theta = 0^\circ$ or $180^\circ$, plane perpendicular to $\vec{B}$)
Potential Energy: — $U = -\vec{M} \cdot \vec{B} = -MB \cos\theta$
Stable Equilibrium: — $\theta = 0^\circ$, $U = -MB$ (minimum potential energy, $\vec{M} \parallel \vec{B}$)
Unstable Equilibrium: — $\theta = 180^\circ$, $U = +MB$ (maximum potential energy, $\vec{M} \uparrow\downarrow \vec{B}$)
Net Force in Uniform B: — $\vec{F}_{net} = 0$
Galvanometer Principle: — $NIAB = k\phi$ (for radial field, $\sin\theta = 1$)

2-Minute Revision

The torque on a current-carrying loop in a uniform magnetic field is a crucial concept. While the net force on such a loop is zero, the forces on its individual segments can create a turning effect, or torque.

This torque is best described using the magnetic dipole moment $\vec{M}$ of the loop, defined as $\vec{M} = NIA \hat{n}$, where $N$ is the number of turns, $I$ is the current, $A$ is the loop's area, and $\hat{n}$ is the unit vector normal to the loop's plane (direction given by the right-hand rule).

The torque $\vec{\tau}$ is then given by the vector cross product $\vec{\tau} = \vec{M} \times \vec{B}$. Its magnitude is $\tau = NIAB \sin\theta$, where $\theta$ is the angle between $\vec{M}$ and the magnetic field $\vec{B}$.

It's vital to remember that $\theta$ is *not* the angle between the plane of the loop and the field, but rather between the normal to the plane and the field. Torque is maximum when $\vec{M}$ is perpendicular to $\vec{B}$ (plane parallel to $\vec{B}$) and zero when $\vec{M}$ is parallel or anti-parallel to $\vec{B}$ (plane perpendicular to $\vec{B}$).

The torque always tries to align $\vec{M}$ with $\vec{B}$. This principle is fundamental to electric motors and moving coil galvanometers, where the magnetic torque is balanced by a restoring torque.

5-Minute Revision

Let's consolidate the concept of torque on a current loop. When a current-carrying loop is placed in a uniform magnetic field, it experiences a rotational effect called torque. This happens because the magnetic forces acting on different sides of the loop, while summing to zero net force (in a uniform field), are not collinear, thus forming a couple. The key to understanding and calculating this torque lies in the magnetic dipole moment, $\vec{M}$.

**1. Magnetic Dipole Moment ($\vec{M}$):** For a loop with $N$ turns, carrying current $I$, and enclosing area $A$, its magnetic dipole moment has a magnitude $M = NIA$. Its direction is perpendicular to the loop's plane, determined by the right-hand rule (curl fingers in current direction, thumb points to $\vec{M}$). For example, a circular coil of radius $r$ has area $A = \pi r^2$, so $M = NI\pi r^2$.

2. Torque Formula: The torque $\vec{\tau}$ experienced by the loop in a magnetic field $\vec{B}$ is given by the vector cross product:

$$\vec{\tau} = \vec{M} \times \vec{B}$$

The magnitude of this torque is $\tau = MB \sin\theta = NIAB \sin\theta$.

Here, $\theta$ is the angle between the magnetic dipole moment vector $\vec{M}$ (normal to the loop's plane) and the magnetic field vector $\vec{B}$. A common mistake is to use the angle between the plane of the loop and the field; if this angle is $\alpha$, then $\theta = 90^\circ - \alpha$.

Example: A coil with $N=20$ turns, area $A=100,\text{cm}^2 = 0.01,\text{m}^2$, current $I=2,\text{A}$, is in a field $B=0.1,\text{T}$. If its plane is at $30^\circ$ to the field, what is the torque? Here, $\alpha = 30^\circ$, so $\theta = 90^\circ - 30^\circ = 60^\circ$. $\tau = (20)(2)(0.01)(0.1) \sin(60^\circ) = 0.04 \times 0.866 = 0.03464,\text{N m}$.

3. Equilibrium and Potential Energy: The potential energy $U$ of a magnetic dipole in a magnetic field is $U = -\vec{M} \cdot \vec{B} = -MB \cos\theta$.

Stable Equilibrium: — Occurs when $\theta = 0^\circ$ ($\vec{M} \parallel \vec{B}$), where $U = -MB$ (minimum potential energy) and $\tau = 0$. The loop's plane is perpendicular to $\vec{B}$.
Unstable Equilibrium: — Occurs when $\theta = 180^\circ$ ($\vec{M}$ anti-parallel to $\vec{B}$), where $U = +MB$ (maximum potential energy) and $\tau = 0$. The loop's plane is also perpendicular to $\vec{B}$, but if slightly perturbed, it will rotate to stable equilibrium.
Maximum Torque: — Occurs when $\theta = 90^\circ$ ($\vec{M} \perp \vec{B}$), where $\tau = NIAB$. The loop's plane is parallel to $\vec{B}$.

4. Applications: This principle is the basis for electric motors (continuous rotation by reversing current using a commutator) and moving coil galvanometers (where magnetic torque is balanced by a spring's restoring torque, $NIAB = k\phi$, with $\sin\theta=1$ due to radial field).

Prelims Revision Notes

Torque on Current Loop: NEET Revision Notes

1. Fundamental Principle:

A current-carrying conductor in a magnetic field experiences a force (Lorentz force). For a closed loop in a uniform magnetic field, the net force is zero, but a net torque can exist.
This torque tends to rotate the loop, aligning its magnetic dipole moment with the external magnetic field.

2. Magnetic Dipole Moment ($\vec{M}$):

Definition: — A vector quantity representing the magnetic strength and orientation of a current loop.
Magnitude: — $M = NIA$

* $N$: Number of turns in the coil. * $I$: Current flowing through the loop (in Amperes). * $A$: Area enclosed by the loop (in $\text{m}^2$). For a circular loop, $A = \pi r^2$; for a rectangular loop, $A = L \times b$.

Direction: — Perpendicular to the plane of the loop, given by the right-hand rule (curl fingers in current direction, thumb points to $\vec{M}$). This is the direction of the normal vector $\hat{n}$.
Units: — Ampere-meter squared ($\text{A m}^2$).

3. Torque ($\vec{\tau}$):

Vector Form: — $\vec{\tau} = \vec{M} \times \vec{B}$
Magnitude: — $\tau = MB \sin\theta = NIAB \sin\theta$

* $B$: Magnetic field strength (in Tesla). * $\theta$: CRITICAL: Angle between the magnetic dipole moment vector $\vec{M}$ (normal to the loop's plane) and the magnetic field vector $\vec{B}$. * Common Trap: If the angle between the *plane* of the loop and $\vec{B}$ is $\alpha$, then $\theta = 90^\circ - \alpha$.

Units: — Newton-meter ($\text{N m}$).

4. Special Orientations:

Maximum Torque: — $\tau_{max} = NIAB$ (when $\theta = 90^\circ$). This occurs when the plane of the loop is parallel to the magnetic field.
Zero Torque: — $\tau = 0$ (when $\theta = 0^\circ$ or $180^\circ$). This occurs when the plane of the loop is perpendicular to the magnetic field.

5. Potential Energy (U):

Formula: — $U = -\vec{M} \cdot \vec{B} = -MB \cos\theta$
Stable Equilibrium: — $\theta = 0^\circ$ ($\vec{M} \parallel \vec{B}$). $U_{min} = -MB$. Loop's plane perpendicular to $\vec{B}$.
Unstable Equilibrium: — $\theta = 180^\circ$ ($\vec{M}$ anti-parallel to $\vec{B}$). $U_{max} = +MB$. Loop's plane perpendicular to $\vec{B}$.
Reference Point: — $U=0$ when $\theta = 90^\circ$ ($\vec{M} \perp \vec{B}$). Loop's plane parallel to $\vec{B}$.

6. Moving Coil Galvanometer:

Principle: — Magnetic torque on the coil ($NIAB$) is balanced by the restoring torque of the suspension wire ($k\phi$).
Radial Field: — In a radial magnetic field, the field lines are always perpendicular to the plane of the coil's sides, ensuring $\theta = 90^\circ$ (or $\sin\theta = 1$) for any deflection. Thus, $\tau = NIAB$.
Equation: — $NIAB = k\phi$

* $k$: Torsional constant of the suspension wire (in $\text{N m/rad}$). * $\phi$: Deflection angle (MUST be in radians).

7. Key Distinctions:

Uniform B-field: — Net force = 0, Torque can be non-zero.
Non-uniform B-field: — Both net force and torque can be non-zero.

Remember: Always convert units to SI and angles to radians for calculations. Pay close attention to the definition of the angle $\theta$ in the torque formula.

Vyyuha Quick Recall

To remember the torque formula and angle: 'M-B-Sin-Theta, Normal-to-Plane is Theta'

M-B-Sin-Theta: — Reminds you $\tau = MB \sin\theta$.
Normal-to-Plane is Theta: — Emphasizes that $\theta$ is the angle between the magnetic moment (which is normal to the plane) and the magnetic field, not the plane itself.