Understanding the Moving Coil Galvanometer: Principles and Applications

The moving coil galvanometer is an instrument used for the detection and measurement of small electric currents through deflection of a coil suspended in a magnetic field. It forms the foundational principle in designing sensitive electrical measuring devices.

Construction of the Moving Coil Galvanometer and Mechanism of Deflection

A moving coil galvanometer consists of a rectangular coil of thin insulated copper wire wound on a light aluminium frame. This coil is suspended so that it can rotate freely about an axis perpendicular to a uniform radial magnetic field provided by a strong cylindrical permanent magnet. The ends of the coil are connected to a circuit externally via flexible leads and a pair of spring-loaded connections that produce a restoring torque. A soft iron cylinder is placed inside the coil to enhance the field's uniformity and intensity.

When an electric current passes through the coil, the interaction between the magnetic field and the current-carrying coil results in a torque. This torque causes the coil to rotate, and the angle of rotation is indicated by a pointer attached to the coil over a calibrated scale.

Principle Underlying the Moving Coil Galvanometer

The working of a moving coil galvanometer is based on the fact that a current-carrying coil placed in a uniform magnetic field experiences a torque which is directly proportional to the current flowing through it. This is a direct application of the Lorentz force law.

Quantitative Analysis of Torque Acting on the Coil

Let a rectangular coil have $N$ turns, each of area $A$, placed in a uniform magnetic field $\vec{B}$. If $I$ is the current flowing through the coil, the force on a length $l$ of wire carrying current $I$ in the magnetic field $B$ directed perpendicularly to $l$ is given by $F = I l B$.

Take the coil to have length $l$ and width $b$. So, the area of each turn $A = l \times b$. The forces on the two vertical sides (length $l$) are equal in magnitude but opposite in direction, producing a couple.

The torque $\tau$ produced by this couple on one turn is given by:

$\tau = \text{Force} \times \text{perpendicular distance}$

The force on each vertical side is $F = I l B$, and the perpendicular distance between forces is $b$. Therefore, the torque for one turn is:

$\tau_{\text{one turn}} = I l B \cdot b$

Since $A = l \cdot b$, rewrite the expression as:

$\tau_{\text{one turn}} = I B A$

For a coil with $N$ turns, the total torque is the sum across all turns:

$\tau = N I B A$

Restoring Torque Due to Suspension and Equilibrium Condition

The coil is suspended by a phosphor bronze strip or a spring, which provides a restoring torque proportional to the angle of deflection $\theta$ from its equilibrium position. If $k$ is the torsional constant of the suspension, the restoring torque is:

$\tau_{\text{restoring}} = k \theta$

At equilibrium, the deflecting torque is balanced by the restoring torque:

$N B A I = k \theta$

Derivation of the Moving Coil Galvanometer Sensitivity Formula

From the equilibrium condition,

$N B A I = k \theta$

Divide both sides by $k$:

$\dfrac{N B A}{k} I = \theta$

The current $I$ in terms of deflection $\theta$ is,

$I = \dfrac{k}{N B A} \theta$

Galvanometer Constant: The term $\dfrac{k}{N B A}$ is called the galvanometer constant. It is the current required for unit deflection in the galvanometer scale.

Current Sensitivity: The current sensitivity of a moving coil galvanometer is defined as the deflection per unit current, i.e., $\dfrac{\theta}{I} = \dfrac{N B A}{k}$.

Conditions for Maximum Sensitivity in a Moving Coil Galvanometer

Current sensitivity increases when the number of turns $N$, the magnetic field $B$, and the area of the coil $A$ are made as large as possible, and the torsional constant $k$ is made as small as possible. Larger $N$, stronger $B$, and larger $A$ increase the torque for a given current, while a smaller $k$ means less restoring force opposes the deflection.

Explanation of the Effect of Shunt Resistance

A low resistance called a shunt is used in parallel with the galvanometer coil to increase the current measuring range. The shunt provides an alternate path for the main current, so only a fraction of the total current passes through the galvanometer. If $S$ is the shunt resistance and $G$ is the resistance of the galvanometer, and $I$ is the total current, the distribution of currents across the galvanometer $I_g$ and shunt $I_s$ satisfy:

$I = I_g + I_s$

Potential difference across $G$ and $S$ (in parallel) is equal, so:

$I_g G = I_s S$

Express $I_s$ in terms of $I$ and $I_g$:

$I_s = I - I_g$

Substitute into the above:

$I_g G = (I - I_g) S$

Expand the equation:

$I_g G = I S - I_g S$

Bring like $I_g$ terms together:

$I_g G + I_g S = I S$

Factor $I_g$:

$I_g (G + S) = I S$

Therefore, the current through the galvanometer is:

$I_g = \dfrac{I S}{G + S}$

Working of the Moving Coil Galvanometer in Quantitative Terms

A moving coil galvanometer works directly according to the equation $I = \dfrac{k}{N B A} \theta$. A given angular deflection $\theta$ on the scale corresponds to a particular value of current $I$, determined by the instrument’s construction parameters and calibration. This enables the galvanometer to serve as a fundamental device for the measurement, detection, and quantification of small currents in an electrical circuit.

Key Features of the Moving Coil Galvanometer and Its Place in the Syllabus

The moving coil galvanometer, addressed in Class 12 Physics (Electromagnetic Induction and Alternating Currents), is essential for understanding the conversion of electrical energy into mechanical effects and is pivotal in laboratory experiments and Class 12 project work. Its mathematical treatment forms the basis for design and analysis of more advanced devices like ammeters and voltmeters, which use the principles and modifications of this fundamental device.