Magnetic Force Formula

The Formula

F = qvB\sin\theta (on a charge) or F = BIL\sin\theta (on a wire of length L).

When to use: A moving charge in a magnetic field feels a sideways push — perpendicular to both its motion and the field. It's like a cross-wind deflecting a moving ball.

Quick Example

A current-carrying wire between two magnets jumps sideways — this is how electric motors work.

Notation

q is the charge in coulombs, \vec{v} is the velocity vector in m/s, \vec{B} is the magnetic field in tesla (T), I is the current in amperes, and L is the wire length in metres. The cross product \times gives a vector perpendicular to both inputs.

What This Formula Means

The force exerted on a moving charge or current-carrying conductor by a magnetic field.

A moving charge in a magnetic field feels a sideways push — perpendicular to both its motion and the field. It's like a cross-wind deflecting a moving ball.

Formal View

The magnetic force on a point charge moving with velocity \vec{v} in a field \vec{B} is given by the Lorentz force law: \vec{F} = q\vec{v} \times \vec{B}. For a straight current-carrying wire of length L, the force is \vec{F} = I\vec{L} \times \vec{B}.

Worked Examples

Example 1

easy
A proton (q = 1.6 \times 10^{-19} \text{ C}) moves at 5 \times 10^6 \text{ m/s} perpendicular to a 0.3 \text{ T} magnetic field. What is the magnetic force on the proton?

Solution

  1. 1
    Use F = qvB\sin\theta with \theta = 90°.
  2. 2
    F = 1.6 \times 10^{-19} \times 5 \times 10^6 \times 0.3 \times 1
  3. 3
    F = 2.4 \times 10^{-13} \text{ N}

Answer

F = 2.4 \times 10^{-13} \text{ N}
The magnetic force on a moving charged particle is perpendicular to both the velocity and the magnetic field. It causes the particle to move in a circular path rather than speeding it up or slowing it down.

Example 2

medium
A wire carrying 8 \text{ A} is 0.5 \text{ m} long and placed at 60° to a 0.4 \text{ T} magnetic field. What force acts on the wire?

Example 3

medium
A proton (q = 1.6 \times 10^{-19} \text{ C}) moves at 3 \times 10^6 \text{ m/s} perpendicular to a 0.5 \text{ T} magnetic field. Find the magnetic force.

Common Mistakes

  • Using the wrong angle — \theta is the angle between the velocity vector and the magnetic field, not between the force and the field.
  • Forgetting that the magnetic force is zero when the charge moves parallel to the field (\sin 0° = 0).
  • Applying the right-hand rule incorrectly for negative charges — the force direction reverses for electrons compared to positive charges.

Why This Formula Matters

The magnetic force on currents is the operating principle of every electric motor.

Frequently Asked Questions

What is the Magnetic Force formula?

The force exerted on a moving charge or current-carrying conductor by a magnetic field.

How do you use the Magnetic Force formula?

A moving charge in a magnetic field feels a sideways push — perpendicular to both its motion and the field. It's like a cross-wind deflecting a moving ball.

What do the symbols mean in the Magnetic Force formula?

q is the charge in coulombs, \vec{v} is the velocity vector in m/s, \vec{B} is the magnetic field in tesla (T), I is the current in amperes, and L is the wire length in metres. The cross product \times gives a vector perpendicular to both inputs.

Why is the Magnetic Force formula important in Physics?

The magnetic force on currents is the operating principle of every electric motor.

What do students get wrong about Magnetic Force?

The force is zero when the charge moves parallel to the field — it's maximum when perpendicular.

What should I learn before the Magnetic Force formula?

Before studying the Magnetic Force formula, you should understand: magnetic field, electric current, force.

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