Question

Explain the principle, construction, and derive the current sensitivity of a moving coil galvanometer.

Hint:

Key results: - Torque on coil: \( \tau = NBAI \) - Current sensitivity: \( \frac{\theta}{I} = \frac{NBA}{k} \) - Radial magnetic field ensures linear scale and high sensitivity.

Answer 1

Principle of Moving Coil Galvanometer
A moving coil galvanometer works on the principle that when a current-carrying coil is placed in a magnetic field, it experiences a torque. This torque causes the coil to rotate. The amount of rotation of the coil is proportional to the current flowing through it. Therefore, a moving coil galvanometer is used to detect and measure small electric currents in a circuit.

Construction of Moving Coil Galvanometer
A moving coil galvanometer consists of a rectangular coil of insulated copper wire wound over a light metallic frame. The coil is placed between the poles of a strong permanent magnet which produces a uniform radial magnetic field. A soft iron cylindrical core is placed inside the coil to make the magnetic field strong and radial.

The coil is suspended by a fine phosphor bronze strip or a torsion wire which also acts as a current lead. When current flows through the coil, it experiences a magnetic torque and rotates. A small mirror or pointer attached to the coil indicates the deflection on a calibrated scale. The restoring torque is provided by the suspension wire, which tries to bring the coil back to its original position.

Working of the Galvanometer
When current flows through the coil placed in the magnetic field, the sides of the coil experience forces due to the magnetic field. These forces produce a torque that rotates the coil. The coil continues to rotate until the magnetic torque is balanced by the restoring torque of the suspension wire.

Derivation of Current Sensitivity
Let the number of turns in the coil be \(N\), the area of the coil be \(A\), the magnetic field strength be \(B\), and the current flowing through the coil be \(I\). The torque acting on the coil due to the magnetic field is given by
\[ \tau = N B I A \] When the coil rotates through an angle \( \theta \), the suspension wire produces a restoring torque proportional to the angle of deflection:
\[ \tau = C\theta \] where \(C\) is the torsional constant of the suspension wire.

At equilibrium, the magnetic torque is equal to the restoring torque:
\[ N B I A = C\theta \] Solving for current \(I\):
\[ I = \frac{C}{N B A}\theta \] Current Sensitivity
Current sensitivity of a galvanometer is defined as the deflection produced per unit current. Therefore,
\[ \text{Current Sensitivity} = \frac{\theta}{I} \] From the above equation,
\[ \frac{\theta}{I} = \frac{N B A}{C} \] Thus, the current sensitivity of a moving coil galvanometer is directly proportional to the number of turns of the coil, the magnetic field strength, and the area of the coil, and inversely proportional to the torsional constant of the suspension wire.

Conclusion
A moving coil galvanometer detects small electric currents based on the torque experienced by a current-carrying coil in a magnetic field. Its current sensitivity is given by
\[ \frac{\theta}{I} = \frac{N B A}{C} \] which shows that increasing the number of turns, magnetic field strength, or coil area increases the sensitivity of the galvanometer.