A moving coil galvanometer experiences a torque = ki, where i is current. If N coils of area A each

Q: Why is the angle between the magnetic field and the area vector always $90^\circ$?

In a moving coil galvanometer, the magnetic poles are made concave and a soft iron core is placed inside the coil. This creates a **radial magnetic field**, ensuring that the plane of the coil is always parallel to the magnetic field lines, maximizing the torque.

Q: What does the torsional constant $C$ represent physically?

The torsional constant $C$ (or $k$ in some textbooks) represents the restoring torque developed per unit angle of twist in the suspension wire. A stiffer wire has a higher $C$.

Q: Why do we use angular impulse instead of just forces for part (c)?

Because the charge $Q$ is passed "suddenly", the current flows for an infinitesimally small time. We cannot track the instantaneous torque, but we can integrate it over time to find the total angular impulse, which directly gives the change in angular momentum.

Q: Why is the kinetic energy converted entirely into potential energy?

As the coil swings, the suspension wire twists, building up elastic potential energy. At the point of maximum deflection ($\theta_{\max}$), the coil momentarily stops, meaning all its initial rotational kinetic energy has been stored as potential energy in the twisted wire.

Q: The official answer key for part (c) doesn't have the current $i$ in the denominator. Is it a typo?

Yes, it is a dimensional and algebraic typo in the original source. If you substitute the torsional constant $C = \frac{2NABi}{\pi}$ into the energy equation, the $i$ must remain in the denominator. The correct expression is $\theta_{\max} = Q\sqrt{\frac{NAB\pi}{2Ii}}$.

Question

A moving coil galvanometer experiences a torque $= ki$, where $i$ is current. If $N$ coils of area $A$ each and moment of inertia $I$ is kept in magnetic field $B$. (2005)
(a) Find $k$ in terms of given parameters.
(b) If for current $i$ deflection is $\frac{\pi}{2}$, find out torsional constant of spring.
(c) If a charge $Q$ is passed suddenly through the galvanometer, find out maximum angle of deflection.

Sigma Percentile · Accepted Answer

The moving coil galvanometer is a beautiful intersection of electromagnetism and mechanics. In this problem, we are going to dissect its behavior under steady currents and sudden jolts of charge. Let's break it down step by step!

### The Anatomy of a Galvanometer

When a current $i$ flows through a coil of $N$ turns and area $A$, placed in a magnetic field $B$, it experiences a magnetic torque. In a well-designed galvanometer, the magnetic poles are shaped to produce a **radial magnetic field**. This ensures that the plane of the coil is always parallel to the magnetic field lines, maximizing the torque at any angle.

The formula for this maximum magnetic torque is:

$$ 	au_m = NABi $$

### Uncovering the Torque Constant

The problem states that the galvanometer experiences a torque given by $	au = ki$. This $k$ is simply a proportionality constant between the torque and the current.

By equating our theoretical torque with the given expression, we can easily find $k$:

$$ ki = NABi $$

$$ k = NAB $$

This perfectly answers part (a) of our problem!

### The Spring's Resistance

As the magnetic torque forces the coil to rotate, the suspension spring at the top begins to twist. This twisting action generates a **restoring torque** that opposes the magnetic torque. The restoring torque is proportional to the angle of twist $	heta$:

$$ 	au_r = C	heta $$

Here, $C$ is the torsional constant of the spring. At equilibrium, the coil stops moving, meaning the magnetic torque is perfectly balanced by the restoring torque:

$$ NABi = C	heta $$

For part (b), we are given a specific calibration condition: for a current $i$, the deflection is exactly $\frac{\pi}{2}$. Let's substitute this into our equilibrium equation to find $C$:

$$ NABi = C \left(\frac{\pi}{2}ight) $$

$$ C = \frac{2NABi}{\pi} $$

This gives us the torsional constant of the spring!

### The Sudden Jolt: Angular Impulse

Part (c) introduces a dynamic scenario. Instead of a steady current, a charge $Q$ is passed *suddenly* through the galvanometer. This means a very large current flows for an infinitesimally small time interval $\Delta t$.

This sudden burst of torque creates an **angular impulse** ($J$), which is the integral of torque over time:

$$ J = \int 	au_m dt = \int (NABi) dt $$

Since $N$, $A$, and $B$ are constants, we can pull them out of the integral:

$$ J = NAB \int i dt $$

We know that the integral of current over time is simply the total charge $Q$. Therefore, the angular impulse is:

$$ J = NABQ $$

This angular impulse acts like a sudden rotational push, changing the angular momentum of the coil from zero to $I\omega$, where $I$ is the moment of inertia and $\omega$ is the initial angular velocity.

$$ I\omega = NABQ $$

$$ \omega = \frac{NABQ}{I} $$

### The Dance of Energy Conservation

Now, the coil is swinging with an initial angular velocity $\omega$. It possesses rotational kinetic energy. As it swings, it twists the spring, converting this kinetic energy into elastic potential energy.

At the point of maximum deflection ($	heta_{\max}$), the coil momentarily comes to rest. By the principle of **conservation of mechanical energy**, the initial kinetic energy is entirely converted into potential energy:

$$ \frac{1}{2}I\omega^2 = \frac{1}{2}C	heta_{\max}^2 $$

Let's rearrange this to solve for $	heta_{\max}$:

$$ 	heta_{\max} = \omega \sqrt{\frac{I}{C}} $$

Now, we substitute the expressions for $\omega$ and $C$ that we derived earlier:

$$ 	heta_{\max} = \left(\frac{NABQ}{I}ight) \sqrt{\frac{I}{\frac{2NABi}{\pi}}} $$

$$ 	heta_{\max} = Q \sqrt{\frac{(NAB)^2 \pi I}{I^2 \cdot 2NABi}} $$

Simplifying the terms inside the square root, we arrive at our final, elegant expression:

$$ 	heta_{\max} = Q \sqrt{\frac{NAB\pi}{2Ii}} $$

*A quick note for the eagle-eyed student: The official answer key omits the current $i$ in the denominator. However, rigorous dimensional analysis and algebraic substitution confirm that the $i$ must be present!*

Visualized Solution

\text{Magnetic Torque on a Coil}

\text{Finding the Constant } k

\text{Restoring Torque and Equilibrium}

\text{Calculating Torsional Constant } C

\text{Sudden Charge Transfer: Angular Impulse}

\text{Initial Angular Velocity}

\text{Energy Conservation for Maximum Deflection}

\text{Final Expression for } \theta_{\max}

The Sigma Insight: Moving Coil Galvanometer

The Anatomy of a Galvanometer

Uncovering the Torque Constant

The Spring's Resistance

The Sudden Jolt

The Dance of Energy Conservation

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