Question

Establish the formula of magnetic dipole moment.

Hint:

The analogy between electrostatics and magnetism is very powerful. Remembering that torque \(\vec{\tau} = \vec{p} \times \vec{E}\) helps in recalling and understanding the magnetic equivalent \(\vec{\tau} = \vec{m} \times \vec{B}\) and the definition of the magnetic dipole moment.

Answer 1

Step 1: Understanding the Concept:
The magnetic dipole moment is a quantity that describes the magnetic strength of a magnet or a current-carrying loop. We can establish its formula by analyzing the torque experienced by a current loop in a uniform magnetic field and drawing an analogy with the torque on an electric dipole.
Step 2: Torque on a Current Loop in a Magnetic Field:
Consider a rectangular loop PQRS with length \( l \) and breadth \( b \), carrying a current I. Let this loop be placed in a uniform magnetic field \(\vec{B}\) such that the normal to the plane of the loop makes an angle \(\theta\) with \(\vec{B}\).
\begin{center} \end{center} The magnetic forces on the arms QR and SP are equal, opposite, and collinear, so they cancel each other out and produce no torque.
The force on arm PQ, using the formula \(\vec{F} = I(\vec{l} \times \vec{B})\), is \(F_{PQ} = I l B\) (directed into the page).
The force on arm RS is \(F_{RS} = I l B\) (directed out of the page).
These two forces, \(F_{PQ}\) and \(F_{RS}\), are equal and opposite, forming a couple that exerts a torque on the loop.
The torque (\(\tau\)) is given by the product of one of the forces and the perpendicular distance between their lines of action. \[ \tau = \text{Force} \times \text{Perpendicular distance} \] The perpendicular distance between the forces is \(b \sin\theta\). \[ \tau = (I l B)(b \sin\theta) \] Since the area of the loop is \(A = l \times b\), we can write: \[ \tau = IAB \sin\theta \] If the loop has N turns, the torque is multiplied by N: \[ \tau = NIAB \sin\theta \] Step 3: Establishing the Formula for Magnetic Dipole Moment:
This expression for torque on a current loop is analogous to the torque experienced by an electric dipole (with dipole moment \(\vec{p}\)) in a uniform electric field (\(\vec{E}\)): \[ \tau_e = pE \sin\theta \] By comparing the two torque equations, we can define a magnetic analogue to the electric dipole moment. This quantity is called the magnetic dipole moment, denoted by m (or \(\mu\)).
Comparing \( \tau = (NIA)B \sin\theta \) with \( \tau_e = pE \sin\theta \), we can establish the formula for the magnitude of the magnetic dipole moment as: \[ m = NIA \] The magnetic dipole moment is a vector quantity, \(\vec{m}\). Its direction is perpendicular to the plane of the current loop, given by the right-hand thumb rule (if the fingers curl in the direction of the current, the thumb points in the direction of \(\vec{m}\)).
Using this vector definition, the torque equation can be written in a more general vector form: \[ \vec{\tau} = \vec{m} \times \vec{B} \] Step 4: Final Answer:
The formula for the magnetic dipole moment of a planar current loop with N turns, area A, and carrying current I is established as \( m = NIA \).