Class 12 > Unit # 18:Magnetic Fields > Force on a Moving Charge in a Magnetic Field

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Unit 18: Magnetic Fields — Force on a Moving Charge in a Magnetic Field

Lecture Overview

Comprehensive lecture notes covering the concept of magnetic force experienced by a moving charge, mathematical derivations, factors affecting it, and how to determine its direction using the right-hand rule.

Source Video: Force on a Moving Charge in a Magnetic Field - Unit 18 Magnetic Fields

1. Introduction & Physical Phenomenon

When charged particles move across a magnetic field, they experience a magnetic force that causes them to deflect. Real-world examples include:

• Television Screen Distortion: Bringing a magnet close to an old CRT TV screen distorts the image and creates unwanted colors due to electron beam deflection.
• Aurora (Northern Lights): Cosmic rays entering Earth's magnetic field experience magnetic forces, bending and creating colorful displays in the northern skies.

Rest vs. Motion: Stationary charges interact only with electric fields (attraction/repulsion), whereas moving charges experience forces from magnetic fields.

2. Experimental Observations

Injecting three different particles into a uniform magnetic field directed perpendicularly into the screen reveals distinct behaviors:

• Neutron (Neutral): Passes straight through un-deflected because it carries no net charge.
• Electron (Negative Charge): Bends downward as it enters the field.
• Proton (Positive Charge): Bends upward as it enters the field.

This confirms that deflection requires both a charge and motion inside a magnetic field.

3. Mathematical Derivation of Force on a Moving Charge

Starting from the magnetic force equation on a current-carrying conductor:

\(F = I L B \sin\theta\) (scalar form)

Using the definition of electric current (\(I = \frac{q}{t}\)), substitute it into the force expression:

\(F = \left(\frac{q}{t}\right) L B \sin\theta = q \left(\frac{L}{t}\right) B \sin\theta\)

Since velocity \(v = \frac{L}{t}\) (distance divided by time), substituting \(v\) yields the final magnitude formula:

\(F = q v B \sin\theta\)

Expressed in vector form using cross-product notation:

\(\vec{F} = q (\vec{v} \times \vec{B})\)

4. Factors Affecting Magnetic Force

The magnitude of the magnetic force depends on four major factors:

• Magnitude of Charge (\(q\)): Greater charge experiences a proportionally stronger magnetic force.
• Velocity of the Particle (\(v\)): If velocity is zero (\(v = 0\)), no magnetic force acts on the charge; only electrostatic forces apply.
• Magnetic Field Strength (\(B\)): Represented by uniform field lines indicating consistent strength and direction.
• Angle (\(\theta\)): The angle between the velocity vector and the magnetic field vector, factored via \(\sin\theta\).

5. Direction of Force: Right-Hand Rule

The direction of the magnetic force on a moving charge is determined using the Right-Hand Rule:

• Thumb: Points in the direction of the particle's velocity (\(\vec{v}\)).
• Fingers: Point in the direction of the magnetic field (\(\vec{B}\)).
• Palm (Force Direction):
  • For Positive Charges (e.g., Protons), the force points outward from the palm.
  • For Negative Charges (e.g., Electrons), the force points into the palm.

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