Moving Coil Galvanometer — Revision Notes

⚡ 30-Second Revision

Principle: — Torque on current loop in B-field: $\tau = NIAB \sin\theta$.
Radial Field: — Ensures $\theta=90^\circ$, so $\tau = NIAB$.
Equilibrium: — $NIAB = k\phi \implies \phi = \frac{NAB}{k}I$.
Current Sensitivity ($I_s$): — $I_s = \frac{\phi}{I} = \frac{NAB}{k}$.
Voltage Sensitivity ($V_s$): — $V_s = \frac{\phi}{V} = \frac{NAB}{kR_g}$.
Ammeter Conversion (Shunt): — $R_{sh} = \frac{I_g R_g}{I - I_g}$ (parallel connection).
Voltmeter Conversion (Series): — $R_{series} = \frac{V}{I_g} - R_g$ (series connection).
Soft Iron Core: — Increases $B$, makes field radial.
Phosphor Bronze: — Low $k$, high elasticity for suspension.
Damping: — Electromagnetic damping by eddy currents in metallic frame.

2-Minute Revision

The Moving Coil Galvanometer (MCG) detects and measures small currents based on the principle that a current-carrying coil experiences a torque in a magnetic field. The torque is given by $\tau = NIAB \sin\theta$.

A crucial design feature is the radial magnetic field, created by concave pole pieces and a soft iron core, which ensures $\sin\theta = 1$, making the torque $\tau = NIAB$ and directly proportional to current.

The soft iron core also concentrates the magnetic field, increasing sensitivity. The coil, suspended by a phosphor bronze wire (chosen for low torsional constant $k$ and high elasticity), deflects until the magnetic torque balances the restoring torque ($k\phi$).

Thus, $\phi = (NAB/k)I$, showing a linear relationship between deflection and current. Current sensitivity ($I_s = NAB/k$) and voltage sensitivity ($V_s = NAB/(kR_g)$) are key metrics. To convert an MCG into an ammeter, a low shunt resistance ($R_{sh} = I_g R_g / (I - I_g)$) is connected in parallel.

For a voltmeter, a high series resistance ($R_{series} = V/I_g - R_g$) is connected in series. Electromagnetic damping, caused by eddy currents in the coil's metallic frame, ensures quick, stable readings.

5-Minute Revision

The Moving Coil Galvanometer (MCG) is an essential instrument for detecting and measuring small electric currents, operating on the principle of torque experienced by a current-carrying coil in a magnetic field. The torque is given by $\tau = NIAB \sin\theta$, where $N$ is the number of turns, $I$ is the current, $A$ is the coil area, $B$ is the magnetic field strength, and $\theta$ is the angle between the normal to the coil and the magnetic field.

Key to MCG design is the use of concave pole pieces of strong permanent magnets and a cylindrical soft iron core. This arrangement creates a radial magnetic field, ensuring that the magnetic field lines are always perpendicular to the plane of the coil's sides.

This makes $\theta = 90^circ$ (or $\sin\theta = 1$) for all deflections, so the torque is always maximum and directly proportional to the current ($\tau = NIAB$). The soft iron core also concentrates the magnetic field, increasing $B$ and thus the galvanometer's sensitivity.

The coil is suspended by a thin phosphor bronze wire (or spring), which provides a restoring torque $\tau_{restoring} = k\phi$, where $k$ is the torsional constant and $\phi$ is the deflection angle. At equilibrium, $NIAB = k\phi$, leading to $\phi = \left(\frac{NAB}{k}\right)I$. This linear relationship allows for a uniform scale.

Sensitivity is crucial:

Current Sensitivity ($I_s$): — $I_s = \frac{\phi}{I} = \frac{NAB}{k}$. To increase $I_s$, increase $N, A, B$ or decrease $k$.
Voltage Sensitivity ($V_s$): — $V_s = \frac{\phi}{V} = \frac{NAB}{kR_g}$. To increase $V_s$, increase $N, A, B$ or decrease $k, R_g$.

Conversion to Ammeter: To measure larger currents, a low resistance called a **shunt resistance ($R_{sh}$)** is connected in parallel with the galvanometer. The formula is $R_{sh} = \frac{I_g R_g}{I - I_g}$, where $I_g$ is the full-scale deflection current of the galvanometer and $R_g$ is its resistance.

Conversion to Voltmeter: To measure larger voltages, a high resistance called a **series resistance ($R_{series}$)** is connected in series with the galvanometer. The formula is $R_{series} = \frac{V}{I_g} - R_g$, where $V$ is the maximum voltage to be measured.

Damping: Electromagnetic damping occurs due to eddy currents induced in the metallic frame of the coil as it moves, quickly bringing the coil to rest without oscillation.

Example: A galvanometer with $R_g = 100\,\Omega$ and $I_g = 1\,\text{mA}$ needs to measure $10\,\text{A}$. Shunt resistance: $R_{sh} = \frac{(10^{-3})(100)}{10 - 10^{-3}} \approx \frac{0.1}{10} = 0.01\,\Omega$. To measure $100\,\text{V}$: Series resistance: $R_{series} = \frac{100}{10^{-3}} - 100 = 100000 - 100 = 99900\,\Omega = 99.9\,\text{k}\Omega$.

Prelims Revision Notes

Principle: — Moving Coil Galvanometer (MCG) works on the principle that a current-carrying coil placed in a magnetic field experiences a torque. This torque is $\tau = NIAB \sin\theta$.
Radial Field: — Essential for MCG. Concave pole pieces and soft iron core create a radial field. This ensures $\theta = 90^circ$ (angle between normal to coil and B-field) at all positions, making $\sin\theta = 1$. Thus, $\tau = NIAB$, ensuring a linear scale.
Soft Iron Core: — Cylindrical, placed inside the coil. Functions: 1. Concentrates magnetic field lines, increasing $B$. 2. Helps in creating a radial magnetic field.
Suspension Wire: — Typically phosphor bronze. Chosen for low torsional constant ($k$) and high elasticity. Provides restoring torque $\tau_{restoring} = k\phi$.
Equilibrium: — Magnetic torque equals restoring torque: $NIAB = k\phi$. This gives deflection $\phi = \left(\frac{NAB}{k}\right)I$.
Current Sensitivity ($I_s$): — Deflection per unit current. $I_s = \frac{\phi}{I} = \frac{NAB}{k}$. Factors increasing $I_s$: increase $N, A, B$; decrease $k$.
Voltage Sensitivity ($V_s$): — Deflection per unit voltage. $V_s = \frac{\phi}{V} = \frac{NAB}{kR_g}$. Factors increasing $V_s$: increase $N, A, B$; decrease $k, R_g$.
Conversion to Ammeter: — Connect a low resistance ($R_{sh}$, shunt) in parallel with the galvanometer. Formula: $R_{sh} = \frac{I_g R_g}{I - I_g}$. Ammeter has low resistance, connected in series.
Conversion to Voltmeter: — Connect a high resistance ($R_{series}$, multiplier) in series with the galvanometer. Formula: $R_{series} = \frac{V}{I_g} - R_g$. Voltmeter has high resistance, connected in parallel.
Damping: — Electromagnetic damping is provided by eddy currents induced in the metallic (e.g., aluminum) frame of the coil, bringing it to rest quickly.
Ideal Ammeter: — Zero resistance.
Ideal Voltmeter: — Infinite resistance.
Units: — Be careful with mA, $\mu\text{A}$, k$\Omega$ conversions in numerical problems.

Vyyuha Quick Recall

N.A.B.K. is SENSITIVE!

N — Number of turns (Increase N, increase sensitivity)
A — Area of coil (Increase A, increase sensitivity)
B — Magnetic field strength (Increase B, increase sensitivity)
K — Torsional constant (Decrease K, increase sensitivity)

Shunt for Ammeter (Parallel, Low R) Series for Voltmeter (Series, High R)