Magnetic Force Formula

Magnetic force is the force exerted on a moving charge or current-carrying conductor by a magnetic field.

The Formula

F=qvBsin⁡θF = qvB\sin\theta (on a charge) or F=BILsin⁡θF = BIL\sin\theta (on a wire of length LL).

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

qq is the charge in coulombs, v⃗\vec{v} is the velocity vector in m/s, B⃗\vec{B} is the magnetic field in tesla (T), II is the current in amperes, and LL 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 v⃗\vec{v} in a field B⃗\vec{B} is given by the Lorentz force law: F⃗=qv⃗×B⃗\vec{F} = q\vec{v} \times \vec{B}. For a straight current-carrying wire of length LL, the force is F⃗=IL⃗×B⃗\vec{F} = I\vec{L} \times \vec{B}.

Worked Examples

Example 1

easy
A proton (q=1.6×10−19 Cq = 1.6 \times 10^{-19} \text{ C}) moves at 5×106 m/s5 \times 10^6 \text{ m/s} perpendicular to a 0.3 T0.3 \text{ T} magnetic field. What is the magnetic force on the proton?

Answer

F=2.4×10−13 NF = 2.4 \times 10^{-13} \text{ N}

First step

1
Use F=qvBsin⁡θF = qvB\sin\theta with θ=90°\theta = 90°.

Full solution

  1. 2
    F=1.6×10−19×5×106×0.3×1F = 1.6 \times 10^{-19} \times 5 \times 10^6 \times 0.3 \times 1
  2. 3
    F=2.4×10−13 NF = 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 A8 \text{ A} is 0.5 m0.5 \text{ m} long and placed at 60°60° to a 0.4 T0.4 \text{ T} magnetic field. What force acts on the wire?

Example 3

medium
A proton (q=1.6×10−19 Cq = 1.6 \times 10^{-19} \text{ C}) moves at 3×106 m/s3 \times 10^6 \text{ m/s} perpendicular to a 0.5 T0.5 \text{ T} magnetic field. Find the magnetic force.

Common Mistakes

  • Using the wrong angle — θ\theta is the angle between the velocity vector and the magnetic field, not between the force and the field. - Fix this by naming the system, checking "Am I using a field or potential to explain how one object influences another across space?", and attaching units or direction to the final statement.
  • Forgetting that the magnetic force is zero when the charge moves parallel to the field (sin⁡0°=0\sin 0° = 0). - Fix this by naming the system, checking "Am I using a field or potential to explain how one object influences another across space?", and attaching units or direction to the final statement.
  • Applying the right-hand rule incorrectly for negative charges — the force direction reverses for electrons compared to positive charges. - Fix this by naming the system, checking "Am I using a field or potential to explain how one object influences another across space?", and attaching units or direction to the final statement.
  • Using magnetic force from a keyword alone - Signal words like field, charge, magnet only point to a possible model; the system must match too.

Why This Formula Matters

Magnetic Force gives students a way to explain non-contact forces and energy changes. It connects electricity, magnetism, gravitation, induction, motors, generators, and orbital motion through a shared spatial model.

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?

qq is the charge in coulombs, v⃗\vec{v} is the velocity vector in m/s, B⃗\vec{B} is the magnetic field in tesla (T), II is the current in amperes, and LL 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?

Magnetic Force gives students a way to explain non-contact forces and energy changes. It connects electricity, magnetism, gravitation, induction, motors, generators, and orbital motion through a shared spatial model.

What do students get wrong about Magnetic Force?

Students often know a formula related to magnetic force but skip the recognition step: Am I using a field or potential to explain how one object influences another across space? That leads to a correct-looking substitution attached to the wrong physical model.

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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