KAHIBARO
Table of Contents
Magnetic force on a wire
A magnetic field can push on moving electric charges. Since an electric current is made of moving charges inside a wire, a wire carrying current can also feel a magnetic force. This effect is one of the most important practical uses of magnetism. It is the basic idea behind electric motors, loudspeakers, and many measuring devices.
To understand this force, imagine a straight wire placed in a magnetic field. If current flows through the wire, and the wire is not parallel to the magnetic field, the wire experiences a force. This force is perpendicular both to the direction of the current and to the magnetic field.
Formula for the magnetic force
For a straight segment of wire of length $L$ in a uniform magnetic field $B$, the magnitude of the magnetic force is
$$
F = I L B \sin \theta
$$
where $I$ is the current in the wire, $L$ is the length of the wire inside the magnetic field, and $\theta$ is the angle between the current direction and the magnetic field.
This formula shows several important facts. The force becomes larger when the current is larger. The force also becomes larger when the magnetic field is stronger. A longer wire in the field feels a larger force. The force depends on angle, and is maximum when the wire is perpendicular to the field, so $\theta = 90^\circ$ and $\sin \theta = 1$. The force is zero when the wire is parallel or anti-parallel to the field, so $\theta = 0^\circ$ or $180^\circ$.
Important formula:
$$
F = I L B \sin \theta
$$
Special cases:
$$
F_{\max} = I L B \quad \text{when } \theta = 90^\circ
$$
$$
F = 0 \quad \text{when } \theta = 0^\circ \text{ or } 180^\circ
$$
Direction of the force
The direction of the force is found using a right-hand rule. Point your fingers in the direction of the current. Then curl or rotate them toward the magnetic field direction. Your thumb points in the direction of the force on the wire.
Because direction matters, the magnetic force is best written in vector form as
$$
\vec{F} = I \vec{L} \times \vec{B}
$$
Here $\vec{L}$ is a vector pointing in the direction of the current, with magnitude equal to the length of the wire segment.
This cross product form explains why the force is perpendicular to both the current and the magnetic field.
Vector form of the magnetic force on a straight wire:
$$
\vec{F} = I \vec{L} \times \vec{B}
$$
The force is perpendicular to both $\vec{L}$ and $\vec{B}$.
Visual picture
Consider a horizontal wire carrying current to the right in a magnetic field pointing upward on the page. The force will point out of the page or into the page depending on the directions chosen. In many textbook examples, the field is drawn with symbols to show direction. Dots usually mean out of the page, and crosses usually mean into the page.
In this drawing, the current points to the right and the magnetic field points upward. The force points downward.
Why the force appears
Inside the wire, many charges move through the conductor. Each moving charge feels a magnetic force. The total force on all these moving charges is transferred to the wire itself, so the whole wire moves.
A full microscopic explanation belongs with the study of charged particles in magnetic fields, but the main idea is simple. Moving charges feel magnetic forces, and a current is moving charge.
Uniform field examples
When the wire is in a uniform field and remains straight, the force is easy to calculate. Some common situations are shown below.
| Situation | Angle $\theta$ | Force magnitude |
|---|---|---|
| Wire parallel to field | $0^\circ$ | $0$ |
| Wire perpendicular to field | $90^\circ$ | $ILB$ |
| Wire at an angle | any $\theta$ | $ILB\sin\theta$ |
Suppose a wire carries a current of $2.0\ \text{A}$, has a length of $0.30\ \text{m}$ in a magnetic field, and the field strength is $0.50\ \text{T}$. If the wire is perpendicular to the field, then
$$
F = ILB = (2.0)(0.30)(0.50) = 0.30\ \text{N}
$$
So the wire feels a force of $0.30\ \text{N}$.
Force on a wire segment in general
The simple formula $F = ILB\sin\theta$ is for a straight wire segment in a uniform field. In a more general case, the magnetic field may vary from one point to another, or the wire may be curved. Then we use a small piece of wire, $d\vec{\ell}$, and write
$$
d\vec{F} = I \, d\vec{\ell} \times \vec{B}
$$
The total force on the wire is found by adding all these small forces:
$$
\vec{F} = I \int d\vec{\ell} \times \vec{B}
$$
For beginners, the straight wire in a uniform field is the most important case.
Opposite current directions
If the current direction is reversed, the force direction also reverses. If the magnetic field direction is reversed, the force direction reverses as well. If both are reversed, the force direction stays the same.
This follows directly from the vector formula.
Changing direction rules:
If $I$ reverses, $\vec{F}$ reverses.
If $\vec{B}$ reverses, $\vec{F}$ reverses.
If both reverse, $\vec{F}$ stays unchanged.
A wire loop in a magnetic field
If a wire is bent into a loop and placed in a magnetic field, different parts of the loop feel forces in different directions. In some cases the total force on the loop is zero, but the forces can still make the loop turn. That turning effect is very important in motors. The turning effect itself is treated separately under torque on current loops, but it starts from the same wire force law.
Practical meaning
The force on a current-carrying wire allows electrical energy to produce motion. This is one of the key links between electricity and mechanics. A wire in a magnetic field can move, rotate, vibrate, or push other objects. In devices such as motors, carefully arranged wires and magnetic fields create controlled motion.
Common mistakes
A very common mistake is to use the full wire length even when only part of the wire is inside the magnetic field. In the formula, $L$ must be the length actually within the field.
Another common mistake is forgetting the angle factor $\sin\theta$. If the wire is not perpendicular to the field, the force is smaller than $ILB$.
Students also often confuse the direction of current with the direction of electron motion. In physics, current direction is defined as the direction positive charge would move. The force formula uses current direction, not electron drift direction.
Be careful:
Use the length of wire inside the magnetic field.
Use the angle between the current direction and the magnetic field.
Use conventional current direction in the right-hand rule.
Summary
A current-carrying wire in a magnetic field experiences a force because moving charges feel magnetic effects. For a straight wire in a uniform magnetic field,
$$
\vec{F} = I \vec{L} \times \vec{B}
$$
and its magnitude is
$$
F = ILB\sin\theta
$$
The force is greatest when the wire is perpendicular to the field and zero when the wire is parallel to the field. The direction is found with the right-hand rule. This simple idea is the foundation for many electromagnetic devices.