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Question

A short bar magnet of magnetic moment $M$ is placed in a non-uniform magnetic field $B$. If the gradient of the field is $\frac{dB}{dx}$ along the axis of the magnet, the force on the magnet is:

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
 **Primary Concept**: Force on a magnetic dipole in a non-uniform magnetic field
 **Related Topics**: Magnetism, magnetic moment, magnetic field gradient
 **Prerequisites**: Understanding of magnetic dipoles, magnetic fields, and calculus (derivatives).
## SOLUTION PATHWAY
**Given Information**:
  * Magnetic moment of the bar magnet: $M$
  * Non-uniform magnetic field: $B$
  * Gradient of the magnetic field along the axis of the magnet: $\frac{dB}{dx}$
**Method & Steps**:
 1. The force on a magnetic dipole in a non-uniform magnetic field is given by the formula:
  $F = M \frac{dB}{dx}$
  Reasoning: The force arises due to the interaction between the magnetic moment of the bar magnet and the spatial variation of the magnetic field.  A stronger field pulls more strongly on the pole closer to it.
  Formula/Rule: $F = M \frac{dB}{dx}$
 2. Substitute the given values into the formula.
  Working:  The force is directly proportional to the magnetic moment ($M$) and the gradient of the magnetic field ($\frac{dB}{dx}$).
  Key Point: The formula is applicable only for a short bar magnet in a non-uniform field. For long magnets or significantly non-uniform fields, a more complex calculation is needed.
 3. The force on the magnet is $M \frac{dB}{dx}$ 
  Result: $M\frac{dB}{dx}$
  Verification: The units are consistent (magnetic moment times field gradient gives a force).
 ## OPTION ANALYSIS
A ✓:  $M\frac{dB}{dx}$: This is the correct formula for the force on a magnetic dipole in a non-uniform magnetic field. 
B :x: $2M\frac{dB}{dx}$: This is incorrect. There's no factor of 2 in the standard formula.
C :x: $\frac{M}{2}\frac{dB}{dx}$: This is incorrect. There's no factor of 1/2 in the standard formula.
D :x: $M^2\frac{dB}{dx}$:  This is incorrect. The force is linearly proportional to the magnetic moment, not its square.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Incorrectly using the formula for a uniform field:  In a uniform field, the net force on a dipole is zero.  Remember this formula only applies to non-uniform fields.
  2.Misinterpreting the gradient: The gradient $\frac{dB}{dx}$ represents the rate of change of the magnetic field strength along the axis of the magnet, not the field itself.
 **Quick Tips**:
  1.Remember the formula $F = M \frac{dB}{dx}$ for the force on a short magnetic dipole in a non-uniform field.
  2.Visualize the stronger field pulling more strongly on the closer pole to understand the direction of the force.

NAGA SAI · Answer

The correct Option is "$M\frac{dB}{dx}$"

NAGA SAI · Answer

$2M\frac{dB}{dx}$

NAGA SAI · Answer

$\frac{M}{2}\frac{dB}{dx}$

NAGA SAI · Answer

$M^2\frac{dB}{dx}$

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