Forces between two parallel wires

Currents generate magnetic fields and also feel forces from magnetic fields. So, two parallel currents (say $I_1$ and $I_2$) will feel forces on each other. To calculate this:

  1. First, notice magnetic field from a wire on itself is zero. The Biot-Savart law involves $\vec{dL} \times \hat{r}$, where $\vec{dL}$ is along the wire and $\hat{r}$ is the displacement vector from the wire to the point where we want to know the field. If the wire is straight, $\vec{dL}$ and $\hat{r}$ are parallel, so the cross product is zero.
  2. Then, calculate the magnetic field from one wire at the position of the other wire. We did this last lecture, and the answer is $\vec{B} = \frac{\mu_0 I_1}{2\pi r} \hat{\phi}$, where $r$ is the distance between the wires and $\hat{\phi}$ is the azimuthal unit vector.
  3. Finally, use $\vec{F} = I_2 \int_L d\vec{L} \times \vec{B}$ to calculate the force on the second wire. Since the wire is straight up the $z$ axis, we can replace the integral with $\vec{F} = I_2 L \,\hat{\vec{k}} \times \vec{B}$, where $L$ is the length of the wire. This gives:
\[\vec{F} = -\frac{\mu_0 I_1 I_2 L}{2\pi r} \hat{\vec{r}}.\]

Note the negative sign. The force is attractive, pulling the wires together, assuming $I_1$ and $I_2$ have the same sign (current flowing in the same direction).

Discussion point: We have used the right-hand-rule to determine the direction of the force. What would happen if we lived in a mirror universe, i.e. where we calculate magnetic fields using a left-hand-rule?

Force on a current loop

The net force on a circular current loop in a uniform magnetic field is always zero.

  • This is because the force on opposite sides of the loop is always equal and opposite (${\rm }d \vec{L}$ is equal and opposite).
  • In the lecture we will look at this more explicitly calculated.
  • Even though the forces balance, a loop can still feel a net torque from a magnetic field, as we will see below.

Torque on a current loop

Calculating the net torque on a current loop is conceptually simplest using maths that many of you won’t hace seen yet (in particular something called Stokes’ theorem).

Using maths you already know it’s still possible to calculate the net torque on a square loop of wire, it’s just annoyingly messy.

Here are the steps (we will follow these in the lectures, and see also p. 925 of YF):

  1. Imagine the square loop of wire is in a magnetic field along the $z$ axis, the loop pivots along the $y$ axis, and makes an angle $\phi$ with the $x$ axis.
  2. Calculate the force on each side of the loop from the magnetic field using $\vec{F} = I \vec{L} \times \vec{B}$, where $\vec{L}$ is the vector displacement along the segment of wire.
  3. Looking at the direction of these forces, note they try to pull the loop into the $x-y$ plane. (You need a sketch to make this remotely intuitive!)
  4. Calculate the net torque on the loop by summing up the torques from each side of the loop, using $\vec{\tau} = \vec{r} \times \vec{F}$.
  5. The end result is that the torque on the square loop is along the $x$ direction, with magnitude $\tau = I A B \sin \phi$, where $A$ is the area of the loop. Note that if $\phi=0$, the loop is in the $x-y$ plane, and there is no torque.
  6. If one introduces an area vector $\vec{A}$ with magnitude $A$ and direction perpendicular to the loop, one can write the torque as $\vec{\tau} = I \vec{A} \times \vec{B}$.
  7. One finally argues that any loop of current, not just this square one we made up, will feel a torque in agreement with this equation, e.g. by imagining the loop as made up of many small square loops where the interior overlapping currents cancel out.

We normally define the magnetic moment as $\vec{\mu} = I \vec{A}$, so the torque on a loop in a magnetic field is

\[\vec{\tau} = \vec{\mu} \times \vec{B}.\]

Discussion point: Can a current loop feel a torque from its own magnetic field?

Potential energy of a magnetic moment

  • Since the magnetic moment of a loop tries to align with the magnetic field, it has a potential energy associated with it, just like an electric dipole in an electric field (see lecture 4).
  • The maths for going from a torque to a potential energy is identical to that done for the electric field in YF Sec 21.7.

  • So the potential energy of the magnetic moment $\vec{\mu}$ in a magnetic field $\vec{B}$ is

    \[U = -\vec{\mu} \cdot \vec{B}.\]
  • Note that, just as in the electric case, the negative sign is because the potential energy is lower when the magnetic moment is aligned with the field.

Magnets as current loops

  • A magnet in the way we usually think of it tries to align itself along magnetic field lines (think e.g. of a compass)…
  • … which is exactly what we just showed will happen with a magnetic moment / current loop!
  • This correctly suggests that a permanent magnet has some kind of magnetic moment inside it.
  • The origin of the moments is complex – a bit about this will be in Lecture 13.
  • The overall magnetic moment is the vector sum of all the individual moments.
  • If the moments are randomly oriented, the magnetic torques cancel out.
  • If the moments are aligned, the net magnetic effects become strong.

Magnetic attraction

Try the example below where the effect of a uniform $\vec{B}$ field on a magnetic dipole is illustrated. The purple currents are “test currents”, i.e. we don’t see the field they generate, just the forces that are acting on them. If you tick/untick the box you will see the net force and torque on the dipole, or the individual forces on the charges.

From playing with this, you should find the dipole tries to align itself with the magnetic field but there is no net force on it. This is because the force on the two charges is equal and opposite, so the net force is zero. So, how does magnetic attraction work, where two magnets stick together?

The answer is that magnetic attraction only works when the field is not uniform. In the next example, the field is generated by a magnetic dipole (the black one), and the test dipole (purple) is attracted to it. Again, compare the individual forces and the net forces/torque. You should be able to explore how current loops can attract or repel each other depending on whether they are aligned or anti-aligned.