In the Class 12 practical, every bench has a galvanometer with a centre-zero scale, often thirty divisions each side. Touch a cell to it through a large resistance and the needle swings to a steady reading. Reverse the cell and it swings the other way. Double the current and the reading doubles. Inside the box there is only a magnet, a coil and a tiny spring.
A coil on a spring in a magnet
A moving-coil galvanometer has a light rectangular coil of many turns, wound on a frame. The coil hangs between the poles of a strong horseshoe magnet, free to turn about a vertical axis. A fine spring or suspension strip twists as the coil turns and pulls it back. The current to be measured flows through the coil. A pointer fixed to the coil moves over the scale.
In a plain uniform field the torque on the coil would change as it turned. So the pole pieces are cut concave, as parts of a cylinder. A cylinder of soft iron sits inside the coil, sharing the same axis. The field lines then cross the narrow gap along radii, like the spokes of a wheel. Such a field is called a radial field.
In the radial field of Fig 6.1, each side of the coil moves along a circle in the gap. The field at a side always points along the radius, so it always lies in the plane of the coil. The forces on the sides stay at right angles to the coil, with the full arm. So the angle between \(\vec M\) and \(\vec B\) is always \(90°\), and the torque is always \(NIAB\). In the uniform field the arm shrinks, and the torque falls to \(NIAB\cos\theta\) at a deflection \(\theta\).
The soft iron core does two jobs. It is easily magnetised, so it gathers the field lines into the narrow gap and makes the field strong. A strong field means a large torque for a small current. And its round surface, with the curved pole faces, makes the field radial.
A moving-coil galvanometer has a radial field. As its coil turns, what happens to the angle between the plane of the coil and the field?
Where the coil stops
The current makes a deflecting torque \(NIAB\). As the coil turns through \(\theta\), the spring twists and pulls back with a restoring torque \(k\theta\). Here \(k\), in N m/rad, is the torsion constant of the spring. The coil turns until the two torques are equal: \(NIAB = k\theta\).
A galvanometer has \(N = 100\), \(A = 2 \times 10^{-4}\,\text{m}^2\), \(B = 0.5\,\text{T}\) and \(k = 10^{-4}\,\text{N m/rad}\). Find the deflection for each milliampere, the current for a deflection of \(0.2\,\text{rad}\), and the deflection for \(0.5\,\text{mA}\).
In the radial field the torques balance: \(NIAB = k\theta\).
\(\theta/I\) \(= NAB/k\) \(= (100 \times 2 \times 10^{-4} \times 0.5)/10^{-4}\) \(= 100\,\text{rad/A}\).
For \(1\,\text{mA}\), \(\theta = 100 \times 10^{-3} = 0.1\,\text{rad}\), about \(5.7°\). For \(0.2\,\text{rad}\), \(I = 0.2/100 = 2\,\text{mA}\). For \(0.5\,\text{mA}\), \(\theta = 0.05\,\text{rad}\).
The deflection is \(0.1\,\text{rad}\) per milliampere, so \(0.2\,\text{rad}\) means \(2\,\text{mA}\) and \(0.5\,\text{mA}\) gives \(0.05\,\text{rad}\).
The spring must be delicate and steady. It is usually a strip or a hair spring of phosphor bronze. This alloy has a small torsion constant, so a tiny torque gives a large deflection. It is very elastic, so it returns to zero and takes no permanent twist. It is non-magnetic, so it does not disturb the field. And it conducts, so it can carry the current into the coil.
A galvanometer coil has \(50\) turns of area \(3 \times 10^{-4}\,\text{m}^2\) in a radial field of \(0.4\,\text{T}\). The torsion constant is \(6 \times 10^{-5}\,\text{N m/rad}\). Find the deflection for a current of \(2\,\text{mA}\), in radians.
How sensitive is it?
A galvanometer is sensitive if a small current gives a large deflection. Its current sensitivity is the deflection per unit current, \(\theta/I = NAB/k\). More turns, a larger coil, a stronger field or a softer spring all raise it. The galvanometer of the stepped example has a current sensitivity of \(100\,\text{rad/A}\), which is \(0.1\,\text{rad}\) per milliampere.
The voltage sensitivity is the deflection per unit potential difference across the coil. The coil has a resistance \(R\), so \(V = IR\) and \(\theta/V = NAB/kR\). Doubling the turns doubles the length of wire, and so doubles \(R\) too. Then the current sensitivity doubles, but the voltage sensitivity stays the same.
In the laboratory a galvanometer is read in divisions of its scale. Its figure of merit is the current needed for a deflection of one division. A galvanometer that shows \(30\) divisions for \(3\,\text{mA}\) has a figure of merit of \(3/30 = 0.1\,\text{mA}\) per division. A smaller figure of merit means a more sensitive instrument.
The galvanometer of the stepped example (\(100\,\text{rad/A}\)) has a coil resistance of \(50\,\Omega\). Find (a) its voltage sensitivity. The coil is then rewound with twice as many turns of the same wire on the same frame. Find (b) the new current and voltage sensitivities.
(a) \(\theta/V = (\theta/I)/R = 100/50 = 2\,\text{rad/V}\).
(b) Twice the turns doubles \(NAB/k\) to \(200\,\text{rad/A}\). The wire is twice as long, so \(R = 100\,\Omega\). Then \(\theta/V = 200/100 = 2\,\text{rad/V}\), unchanged.
A galvanometer shows a full-scale deflection of \(50\) divisions for \(2\,\text{mA}\). What current, in milliamperes, gives a deflection of \(25\) divisions?
The number of turns of a galvanometer coil is doubled, using the same wire. What happens to its voltage sensitivity?
The galvanometer is the heart of the ammeter and the voltmeter. A small resistance joined across it turns it into an ammeter; a large resistance in series turns it into a voltmeter. The chapter Cells, Bridges and Measuring Instruments works out those conversions.
Why the marks are evenly spaced
In the radial field the torque \(NIAB\) does not depend on the angle. So \(\theta = (NAB/k)I\), and the deflection is proportional to the current. Equal steps of current give equal steps of deflection, and the marks on the scale are evenly spaced. This is the gift of the radial field.
In a uniform field the torque would be \(NIAB\cos\theta\) at a deflection \(\theta\). The balance \(k\theta = NIAB\cos\theta\) no longer makes \(\theta\) proportional to \(I\). Each extra milliampere would give a smaller extra deflection than the last. The marks would crowd together at the high end of the scale.
In the sim above, the radial field gives \(14°\) for every milliampere, up to \(43°\) at \(3\,\text{mA}\). In the uniform field the same galvanometer gives \(14°\), \(26°\) and \(35°\) for \(1\), \(2\) and \(3\,\text{mA}\). The steps shrink from \(14°\) to \(12°\) to \(9°\).
A galvanometer is built with flat pole pieces and no iron core, so its field is uniform. How would its scale look?
Early galvanometers moved a magnetic needle near a coil, and the Earth's field upset every reading. In 1882 the French physiologist Jacques-Arsène d'Arsonval turned the idea round. He hung a light coil between the poles of a fixed magnet, so the coil moved and the magnet stayed still. A few years later the American engineer Edward Weston added the curved pole pieces, the iron core and the pivoted pointer. His rugged meter, with its even scale, is the one on the practical bench.
- Writing the deflecting torque as \(NIAB\sin\theta\) in a galvanometer.The radial field keeps the plane of the coil along \(\vec B\), so the torque is \(NIAB\) at every deflection.
- Thinking the iron core only strengthens the field.It also makes the field radial, which makes the scale linear.
- Expecting more turns to raise the voltage sensitivity.More turns raise \(R\) in step, so \(NAB/kR\) stays the same.
- Confusing the figure of merit with the sensitivity.The figure of merit is current per division; a smaller figure of merit means a more sensitive galvanometer.