## Forces on currents in magnetic fields

7-15-99

### The magnetic force on a current-carrying wire

A magnetic field will exert a force on a single moving charge, so it follows that it will also exert a force on a current, which is a collection of moving charges.

The force experienced by a wire of length l carrying a current I in a magnetic field B is given by

Again, the right-hand rule can be used to find the direction of the force. In this case, your thumb points in the direction of the current, your fingers point in the direction of B. Your palm gives the direction of F.

### The force between two parallel wires

Parallel wires carrying currents will exert forces on each other. One wire sets up a magnetic field that influences the other wire, and vice versa. When the current goes the same way in the two wires, the force is attractive. When the currents go opposite ways, the force is repulsive. You should be able to confirm this by looking at the magnetic field set up by one current at the location of the other wire, and by applying the right-hand rule.

Here's the approach. In the picture above, both wires carry current in the same direction. To find the force on wire 1, look first at the magnetic field produced by the current in wire 2. Everywhere to the right of wire 2, the field due to that current is into the page. Everywhere to the left, the field is out of the page. Thus, wire 1 experiences a field that is out of the page.

Now apply the right hand rule to get the direction of the force experienced by wire 1. The current is up (that's your fingers) and the field is out of the page (curl your fingers that way). Your thumb should point right, towards wire 2. The same process can be used to figure out the force on wire 2, which points toward wire 1.

Reversing one of the currents reverses the direction of the forces.

The magnitude of the force in this situation is given by F = IlB. To get the force on wire 1, the current is the current in wire 1. The field comes from the other wire, and is proportional to the current in wire 2. In other words, both currents come into play. Using the expression for the field from a long straight wire, the force is given by:

Note that it is often the force per unit length, F / l, that is asked for rather than the force.

### The torque on a current loop

A very useful effect is the torque exerted on a loop by a magnetic field, which tends to make the loop rotate. Many motors are based on this effect.

The torque on a coil with N turns of area A carrying a current I is given by:

The combination NIA is usually referred to as the magnetic moment of the coil. It is a vector normal (i.e., perpendicular) to the loop. If you curl your fingers in the direction of the current around the loop, your thumb will point in the direction of the magnetic moment.

### Applications of magnetic forces and fields

There are a number of good applications of the principle that a magnetic field exerts a force on a moving charge. One of these is the mass spectrometer : a mass spectrometer separates charged particles (usually ions) based on their mass.

### The mass spectrometer

The mass spectrometer involves three steps. First the ions are accelerated to a particular velocity; then just those ions going a particular velocity are passed through to the third and final stage where the separation based on mass takes place. It's worth looking at all three stages because they all rely on principles we've learned in this course.

#### Step 1 - Acceleration

In physics, we usually talk about charged particles (or ions) being accelerated through a potential difference of so many volts. What this means is that we're applying a voltage across a set of parallel plates, and then injecting the ions at negligible speed into the are between the plates near the plate that has the same sign charge as the ions. The ions will be repelled from that plate, attracted to the other one, and if we cut a hole in the second one they will emerge with a speed that depends on the voltage.

The simplest way to figure out how fast the ions are going is to analyze it in terms of energy. When the ions enter the region between the plates, the ions have negligible kinetic energy, but plenty of potential energy. If the plates have a potential difference of V, the potential energy is simply U = qV. When the ions reach the other plate, all this energy has been converted into kinetic energy, so the speed can be calculated from:

#### Step 2 - the velocity selector

The ions emerge from the acceleration stage with a range of speeds. To distinguish between the ions based on their masses, they must enter the mass separation stage with identical velocities. This is done using a velocity selector, which is designed to allow ions of only a particular velocity to pass through undeflected. Slower ions will generally be deflected one way, while faster ions will deflect another way. The velocity selector uses both an electric field and a magnetic field, with the fields at right angles to each other, as well as to the velocity of the incoming charges.

Let's say the ions are positively charged, and move from left to right across the page. An electric field pointing down the page will tend to deflect the ions down the page with a force of F = qE. Now, add a magnetic field pointing into the page. By the right hand rule, this gives a force of F = qvB which is directed up the page. Note that the magnetic force depends on the velocity, so there will be some particular velocity where the electric force qE and the magnetic force qvB are equal and opposite. Setting the forces equal, qE = qvB, and solving for this velocity gives v = E / B. So, a charge of velocity v = E / B will experience no net force, and will pass through the velocity selector undeflected.

Any charge moving slower than this will have the magnetic force reduced, and will bend in the direction of the electric force. A charge moving faster will have a larger magnetic force, and will bend in the direction of the magnetic force.

A velocity selector works just as well for negative charges, the only difference being that the forces are in the opposite direction to the way they are for positive charges.

#### Step 3 - mass separation

All these ions, with the same charge and velocity, enter the mass separation stage, which is simply a region with a uniform magnetic field at right angles to the velocity of the ions. Such a magnetic field causes the charges to follow circular paths of radius r = mv / qB. The only thing different for these particles is the mass, so the heavier ions travel in a circular path of larger radius than the lighter ones.

The particles are collected after they have traveled half a circle in the mass separator. All the particles enter the mass separator at the same point, so if a particle of mass m1 follows a circular path of radius r1, and a second mass m2 follows a circular path of radius r2, after half a circle they will be separated by the difference between the diameters of the paths after half a circle. The separation is

### The Hall Effect

Another good application of the force exerted by moving charges is the Hall effect. The Hall effect is very interesting, because it is one of the few physics phenomena that tell us that current in wires is made up of negative charges. It is also a common way of measuring the strength of a magnetic field.

Start by picturing a wire of square cross-section, carrying a current out of the page. We want to figure out whether the charges flowing in that wire are positive, and out of the page, or negative, flowing in to the page. There is a uniform magnetic field pointing down the page.

First assume that the current is made up of positive charges flowing out of the page. With a magnetic field down the page, the right-hand rule indicates that these positive charges experience a force to the right. This will deflect the charges to the right, piling up positive charge on the right and leaving a deficit of positive charge (i.e., a net negative charge) on the left. This looks like a set of charged parallel plates, so an electric field pointing from right to left is set up inside the wire by these charges. The field builds up until the force experienced by the charges in this electric field is equal and opposite to the force applied on the charges by the magnetic field.

With an electric field, there is a potential difference across the wire that can be measured with a voltmeter. This is known as the Hall voltage, and in the case of the positive charges, the sign on the Hall voltage would indicate that the right side of the wire is positive.

Now, what if the charges flowing through the wire are really negative, flowing into the page? Applying the right-hand rule indicates a magnetic force pointing right. This tends to pile up negative charges on the right, resulting in a deficit of negative charge (i.e., a net positive charge) on the left. As above, an electric field is the result, but this time it points from left to right. Measuring the Hall voltage this time would indicate that the left side of the wire is negative.

So, the potential difference set up across the wire is of one sign for negative charges, and the other sign for positive charges, allowing us to distinguish between the two, and to tell that when charges flow in wires, they are negative. Note that the electric field, and the Hall voltage, increases as the magnetic field increases, which is why the Hall effect can be used to measure magnetic fields.