Question

A simple galvanometer is made from a coil that consists of \(N\) loops of wire of area \(A .\) The coil is attached to a mass, \(M\), by a light rigid rod of length \(L\). With no current in the coil, the mass hangs straight down, and the coil lies in a horizontal plane. The coil is in a uniform magnetic field of magnitude \(B\) that is oriented horizontally. Calculate the angle from the vertical of the rigid rod as a function of the current, \(i\), in the coil.

Short answer

Question: Determine the angle from the vertical of a rigid rod attached to a mass by considering the torques acting on the system due to gravity and magnetic force. Answer: The angle from the vertical, \(θ\), can be found using the expression: \(θ = \frac{1}{2} arcsin(\frac{2NiAB}{Mg})\), where \(N\) is the number of loops in the coil, \(i\) is the current in the coil, \(A\) is the area of the coil, \(B\) is the magnitude of the magnetic field, \(M\) is the mass, and \(g\) is the acceleration due to gravity.

Step-by-step solution

Identify the forces and torques acting on the system

To find the angle from the vertical, we need to analyze the torques acting on the system. The two forces acting on the system are the gravitational force acting on the mass, and the magnetic force acting on the coil. The gravitational force creates a torque about the axis of rotation that opposes the angle, and the magnetic force creates a torque that helps the angle to increase.

Calculate the torque due to gravitational force

The gravitational force acting on the mass creates a torque about the axis of rotation. This torque can be expressed as: \(τ_g = MgLsin(θ)\) where \(τ_g\) is the torque due to gravity, \(M\) is the mass, \(g\) is the acceleration due to gravity, \(L\) is the length of the rigid rod, and \(θ\) is the angle from the vertical.

Calculate the torque due to magnetic force

The magnetic force acting on the coil creates a torque about the axis of rotation. To calculate this torque, we need to first find the magnetic force acting on each segment of the coil. The magnetic force on a current loop can be expressed as: \(F_m = NiABsin(α)\) where \(F_m\) is the magnetic force, \(N\) is the number of loops in the coil, \(i\) is the current in the coil, \(A\) is the area of the coil, \(B\) is the magnitude of the magnetic field, and \(α\) is the angle between the magnetic field and the plane of the coil. Since the magnetic field is horizontal and the coil lies in a horizontal plane, \(α = 90°\). Therefore, the magnetic force acting on each segment of the coil is: \(F_m = NiAB\) The torque due to this magnetic force can be expressed as: \(τ_m = F_mLcos(θ)\) where \(τ_m\) is the torque due to magnetic force.

Set up the equilibrium conditions and solve for the angle

At equilibrium, the total torque acting on the system is zero. Therefore, we can set up the following equation: \(τ_g = τ_m\) Substituting the values of torques due to gravitational and magnetic forces, we get: \(MgLsin(θ) = NiABLcos(θ)\) Now we need to solve for \(θ\): \(sin(θ)cos(θ)=\frac{NiAB}{Mg}\) Using the trigonometric identity \(2sin(θ)cos(θ)=sin(2θ)\), we can rewrite the equation as: \(sin(2θ) = \frac{2NiAB}{Mg}\) Now solve for \(θ\): \(θ = \frac{1}{2} arcsin(\frac{2NiAB}{Mg})\) This expression gives the angle of the rigid rod from the vertical as a function of the current in the coil.

Key concepts

Magnetic Torque

Magnetic torque is a crucial concept in understanding how galvanometers work. When electric current flows through a coil within a magnetic field, it experiences a force, known as the magnetic force. This force is strongest when the field is perpendicular to the coil's plane, leading to maximum torque. Let’s look into how this works.

In our scenario, the coil with current creates a magnetic moment which interacts with the external magnetic field of magnitude \(B\). This interaction generates a torque \(τ_m\), which can be described mathematically as:

• \(F_m = NiAB\) - where \(F_m\) is the magnetic force, \(N\) is the number of turns in the coil, \(i\) is current, \(A\) is the area, and \(B\) is the magnetic field.
• The torque is given by \(τ_m = F_m L \cos(θ)\), where \(L\) is the distance from the axis, and \(θ\) is the angle from the vertical.

The magnetic torque works to increase the angle \(θ\), which the rod makes with the vertical, opposing gravitational effects. This behavior is key to how measuring devices like galvanometers operate.

Gravitational Torque

When analyzing forces in our galvanometer system, gravitational torque plays a crucial role in determining equilibrium. Gravitational torque opposes the torque generated by the magnetic force, working to maintain a stable system.

Gravitational torque \(τ_g\) can be described by the formula:

• \(τ_g = MgL \sin(θ)\) - where \(M\) is the mass attached to the rod, \(g\) is gravitational acceleration \(9.81 \, m/s²\), \(L\) is the length of the rod from pivot to mass, and \(θ\) is the angle from the vertical.

As the gravitational force pulls directly downward, the rod tends to rotate back towards the vertical position. This opposing torque is crucial when determining the angle \(θ\) at which equilibrium is reached in the presence of the magnetic torque from the coil.

Equilibrium Condition

In our system, equilibrium is a state where the net torque acting on the galvanometer is zero. At this point, the gravitational torque \(τ_g\), which pulls the rod back to a vertical position, exactly balances the magnetic torque \(τ_m\), which shifts it away. Let's explore how this balance achieves harmony.

Equilibrium is mathematically expressed as:

• \(τ_g = τ_m\)
• \(MgL \sin(θ) = NiABL \cos(θ)\)

To find the angle \(θ\), we rearrange the equation using a trigonometric identity, \(2 \sin(θ) \cos(θ) = \sin(2θ)\):

• \(\sin(2θ) = \frac{2NiAB}{Mg}\)
• \(θ = \frac{1}{2} \arcsin(\frac{2NiAB}{Mg})\)

This solution provides the angle \(θ\) that the rod makes with the vertical when the system is in balance, dictating the operational specifics of devices like galvanometers, where such precise control of orientation is essential.