# Lesson Explainer: The Moving-Coil Galvanometer Physics

In this explainer, we will learn how to describe the application of the motor effect to the measuring of electric current by a moving-coil galvanometer.

When a current passes through a wire, a magnetic field is produced around the wire. When this wire is placed in an external magnetic field, the two fields interact, producing a force that is exerted on the wire. If the magnetic field and the current in the wire are perpendicular to each other, the magnitude of this force is given by , where is the magnetic flux density of the external field, is the current in the wire, and is the length of wire in the magnetic field. The direction of this force can be determined using Fleming’s left-hand rule, as shown in the diagram below.

If we have a rectangular loop of current-carrying wire in the magnetic field, the wire loop will experience a torque, which causes it to rotate. This is because the charge is flowing in opposite directions on either side of the loop. Hence, the two sides of the loop experience forces acting in different directions. The rotation of a wire loop in a magnetic field is known as the motor effect, shown in the following diagram.

A galvanometer uses the motor effect to detect and measure a current in an electric circuit. A galvanometer consists of an iron core within a rectangular loop of wire. These are placed within a permanent magnetic field. The magnetic field can be provided by a horseshoe-shaped magnet, where the north and south poles are placed on either side of the components. The north and south poles should be concave, semicircular shapes. The loop of wire is connected to torsion springs by two conducting rods. The torsion springs are also made of a conducting material. They act as the “terminals” of the galvanometer, connecting it to an external electric circuit. The loop of wire is also connected to a pointer, which can move along a scale. The following diagram shows the setup of a galvanometer.

The iron core alters the field of the external magnet. Magnetic field lines always travel from the north pole to the south pole, but because iron is so easily magnetized, it is easier for the field lines of the external magnet to pass through the iron core instead of through the air. This condenses the field lines so that they pass through the coil of wire. This, combined with the concave shape of the magnetic poles, provides a radial magnetic field, as shown in the diagram below. This means that the two sides of the loop that point into and out of the screen are always perpendicular to the external field. Hence, they experience a uniform force, always given by .

When there is a current in the loop of wire, the loop of wire experiences a torque, which causes it to rotate. However, the wire loop does not spin freely. The torsion springs oppose its rotation by applying a torque in the opposite direction. As the loop rotates, the magnitude of the opposing torque from the torsion springs increases. The greater the angle of rotation of the loop, the greater the torque applied by the torsion springs. The size of the torque on the loop of wire due to the magnetic field remains constant. Eventually, the magnitude of the torque from the torsion springs becomes equal to that of the torque due to the motor effect. When this happens, the loop reaches an equilibrium and stops rotating. If the current through the loop is increased, the magnitude of the torque due to the magnetic field also increases. This means the loop will rotate further before the torsion springs fully oppose its motion.

### Example 1: The Motor Effect in Moving-Coil Galvanometers

The diagram shows a moving-coil galvanometer. The terminals of the galvanometer are connected to a direct-current source. Which of terminals A and B connects to the positive output of the source?

We can work out the direction of conventional current (the flow of positive charge) using Fleming’s left-hand rule.

The loop is within a magnetic field, with the south pole on the left and the north pole on the right. Field lines always point from north to south. This means that when we use Fleming’s left-hand rule, the magnetic field, represented by the index finger, points from right to left across the screen.

Let’s consider the edge of the loop on the right-hand side. This part of the loop is experiencing a downward force. In Fleming’s left-hand rule, the force corresponds to the thumb. So, our thumb should be pointing downward.

If we position our left hand like this, with our index finger pointing from right to left and our thumb pointing downward, we find that our middle finger is pointing toward us, out of the screen. The middle finger represents the flow of positive charge.

This is shown in the diagram below.

This tells us the positive charge is flowing from terminal A to terminal B. Hence, terminal A must be connected to the positive output of the source.

The wire loop is connected to a pointer, which moves along a scale as the loop rotates. This allows us to measure the current passing through the loop. When there is no current in the loop, the pointer is in a vertical position, indicating a current of 0 A. When a current is passed through the loop, the loop rotates until it reaches equilibrium with the torsion springs. This causes the pointer to move through some angle, . We call this movement “deflection”. If the magnitude of the current in the loop is increased, the loop will rotate further before it reaches equilibrium, causing the pointer to be deflected through a larger angle.

If the direction of the current is reversed, the direction of the torque due to the magnetic field will also reverse, according to Fleming’s left-hand rule. This means the direction of the loop’s rotation will reverse, and the pointer will be deflected in the opposite direction. So, if the pointer is initially at an angle of to the right, when we reverse the current, the pointer will move to a position at an angle to the left. This means that the galvanometer indicates both the magnitude and the direction of the current. Before it can be used, the galvanometer should be calibrated. This involves passing a current of known magnitude and direction through the galvanometer and measuring its deflection.

Note that the galvanometer can only be used to measure very small currents that have magnitudes in the range of microamperes or milliamperes. If too large a current is passed through the galvanometer, the deflection of the pointer will be too large. This will overstretch the torsion strings, potentially deforming them and causing them to lose their elasticity. This would damage the galvanometer so it could no longer be used to measure current. A large current could also cause the wires in the galvanometer to become too hot. Again, this could damage the galvanometer.

### Example 2: The Current in a Moving-Coil Galvanometer

The diagram shows a moving-coil galvanometer. Full-scale deflection of the galvanometer arm occurs when the galvanometer coils carry a current with a magnitude of 150 μA. Which of the following must be true of the current that passes from contact A to contact B?

1. μA
2. μA

When the current in the loop is 0 μA, the pointer will be in the middle of the scale, pointing vertically upward.

We know that full-scale deflection of the galvanometer arm occurs when the galvanometer coils carry a current with a magnitude of 150 μA. This means that when the coil carries a current of 150 μA, the arm will be deflected to the extreme right of the scale. Similarly, when the coil carries a current of μA, the arm will be deflected to the extreme left of the scale.

In the diagram, the pointer is slightly to the right of the zero point. This means that the coils must be carrying a current that is positive and has a magnitude greater than 0 μA but less than 150 μA.

So, the answer must be D, μA.

### Example 3: The Current in a Moving-Coil Galvanometer

The diagram shows a moving-coil galvanometer. Full-scale deflection of the galvanometer arm occurs when the galvanometer coils carry a current with a magnitude of 150 μA. Which of the following must be true of the current that passes from contact A to contact B?

1. μA
2. μA

When the current in the loop is 0 μA, the pointer will be in the middle of the scale, pointing vertically upward.

We know that full-scale deflection of the galvanometer arm occurs when the galvanometer coils carry a current with a magnitude of 150 μA. This means that when the coil carries a current of 150 μA, the arm will be deflected to the extreme right of the scale. Similarly, when the coil carries a current of μA, the arm will be deflected to the extreme left of the scale.

In the diagram, the pointer is slightly to the left of the zero point. This means that the coils must be carrying a current that is negative and has a magnitude greater than 0 μA but less than 150 μA.

So, the answer must be B, μA.

Galvanometers are built such that the angle of the pointer’s deflection, , is directly proportional to the current, , in the loop, . We can write this as an equation by introducing a constant of proportionality, which we call , for “sensitivity”: , which rearranges to .

Sensitivity has units of degrees per ampere () or radians per ampere (rad/A). The sensitivity tells us about the response of the galvanometer to different magnitudes of current. If the galvanometer has a large sensitivity, then a large angular deflection of the pointer can be caused by a relatively small current. If the sensitivity is small, then even a large current will only create a small angular deflection of the pointer.

### Example 4: The Sensitivity of a Moving-Coil Galvanometer

The arm of a moving-coil galvanometer is deflected through an angle of when the current through the galvanometer is 180 μA. What is the sensitivity of the galvanometer? Answer to two decimal places.

The sensitivity of a galvanometer is defined as the constant of proportionality between the current in the coils and the angle of deflection of the arm, . This rearranges to .

Here, we have and . Therefore,

Notice the units here: because our current was so small, we have chosen to measure the sensitivity in degrees per microampere () rather than in degrees per ampere ().

### Example 5: The Sensitivity of a Moving-Coil Galvanometer

The arm of a moving-coil galvanometer is deflected through an angle of when the current through the galvanometer is 360 μA. The maximum deflection angle for the arm is . What is the maximum value of current that the galvanometer can measure? Answer to the nearest microampere.

In order to calculate the maximum value of current that the galvanometer can measure, we must first calculate the sensitivity of the galvanometer.

We know that the arm is deflected through an angle of when the current through the galvanometer is 360 μA, so we can use these values to calculate the sensitivity:

We also know that the maximum deflection angle for the arm is . We can use the sensitivity to calculate the current that corresponds to a deflection of this angle:

So, the maximum current the coils can carry, which corresponds to the maximum deflection of the arm, is 736 μA.

### Key Points

• A galvanometer uses the motor effect to detect and measure a current in a circuit.
• When a current is passed through a loop of wire in a magnetic field, the loop rotates. In a galvanometer, this rotation causes a pointer to be deflected across a scale.
• The rotation of the loop is opposed by a torque produced by torsion springs. The point at which the system reaches equilibrium indicates the magnitude and direction of current in the wire loop.
• The sensitivity of a galvanometer is defined by and is measured in degrees per ampere ().
• The sensitivity of a galvanometer tells us how large a current is needed to cause a certain change in the angular deflection of the pointer.
• Galvanometers can only be used to measure very small current. Large currents can damage a galvanometer.