Magnetic Dipole Moment- Torque on Current Carrying Loop

What is magnetic dipole moment in physics?

The magnetic dipole moment (µ) is a vector defined as multiplication of current and area vector whose direction is direction of area vector. Direction of area vector is determined by the right hand rule. Magnetic moment is also a name of magnetic dipole moment.

Right hand thumb rule

Firstly, place your hand such that your thumb is perpendicular to plane of loop. Now, turn your fingers in the direction of current. Direction of thumb gives direction of resultant vector.

Magnetic dipole moment formula

Formula of magnetic moment is, µ = iA, where i is current in the loop and A is area vector of that loop.

Unit of dipole moment

SI unit of magnetic dipole moment = unit of i × unit of A = A m².

Direction of dipole moment

Direction of dipole moment is same as direction of area vector.

Magnetic Dipole Moment of a Revolving Electron

The electron of charge (–e) (e = + 1.6 × 10^-19 C) performs uniform circular motion around a stationary heavy nucleus of charge +Ze. We can write equivalent current I for it

I=e/T, where T is time period.

Let r be orbital radius and v be velocity.

⇒ T = 2πr/v ⇒ I = ev/2πr

Now, we know that magnetic dipole moment = µ = iA

⇒ µ = A × ev/2πr

⇒ µ = πr² × ev/2πr = evr/2

Multiplying and dividing by m_e, mass of electron

⇒ µ = evrm_e / 2m_e

but m_evr is angular momentum, l

⇒ µ = el / 2m_e

but µ = -el / 2m_e

The negative sign indicates that the angular momentum of the electron is opposite in direction to the magnetic moment.

Gyromagnetic Ratio

Gyromagnetic ratio the ratio of magnetic dipole moment due to orbiting electron to its orbital angular momentum.

Thus, Gyromagnetic ratio = µ/l = e/2m_e

Torque on Current Loop in Uniform Magnetic Field

Let’s consider a rectangular loop placed such that the uniform magnetic field B is in the plane of the loop as shown. Current in loop is I.

torque on a rectangular current loop in a uniform

The magnetic field exerts no force on the two arms AD and BC of the loop because current in it makes 0° or 180° with magnetic field.

Since, magnetic field is perpendicular to the arm AB of the loop, it exerts a force F₁ on it. F₁ is directed into the plane of the loop.

Thus, F₁ = BIb

Similarly, let force on arm CD be F₂.

So, F₂ = BIb

Thus, torque = τ = F₁a/2 + F₂a/2

⇒ τ = ^BIba/₂ + ^BIba/₂ = BI(ab)

⇒ τ = BIA, where A = ab is the area of rectangular loop.

When loop turns angle θ, τ = BIA sinθ.

Vector form of torque = τ = I(A × B), where A and B are vectors.

But we know that, IA=μ

⇒ τ = μ × B

Thus, magnetic dipole moment can also be defined as, a vector quantity which when cross multiplied by magnetic field vector gives torque.

Circular Current Loop As a Magnetic Dipole

Here, we will see that the magnetic field (at large distances) due to current in a circular current loop is very similar in behavior to the electric field of an electric dipole.

Let us consider a circular loop of radius R carrying current I.

Thus, magnetic field on the axis of loop at distance x is

magnetic field at centre of circular loop

Note: if you don’t know how this formula came, then visit this blog.

When, x>>R, B = μ₀IA/2πx³

Since, μ=IA

⇒ B = μ₀2m/4πx³