Question

A tangent galvanometer is an instrument, developed in the ninetecnth century, designed to measure current based on the deflection of a compass needle. A coil of wire in a vertical plane is aligned in the magnetic north-south direction. A compass is placed in a horizontal plane at the center of the coil. When no current flows, the compass necdle points directly toward the north side of the coil. When a current is sent through the coil, the compass needle rotates through an angle \(\theta\). Derive an equation for \(\theta\) in terms of the number of coil turns \(N,\) the coil radius \(r,\) the coil current \(I,\) and the horizontal component of Earth's field \(B_{\mathrm{H}} .\) [Hint: The name of the instrument is a clue to the result. \(]\) (IMAGE CANNOT COPY)

Short answer

The deflection angle \(\theta\) is given by \(\text{tan}(\theta) = \frac{\mu_0 \cdot N \cdot I}{2rB_H}\).

Step-by-step solution

Understand the Physics

The tangent galvanometer employs the principle of balancing magnetic forces. When current flows through the coil, it creates a magnetic field that deflects the needle. This deflection is measured by the angle \(\theta\). The magnetic field from the coil \(B_C\) and Earth's magnetic field \(B_H\) effect the needle's position.

Define the Coil's Magnetic Field

The magnetic field produced by the coil of radius \(r\) and \(N\) turns, carrying a current \(I\), is given by the formula: \[ B_C = \frac{\mu_0 \cdot N \cdot I}{2 \cdot r} \] Here \(\mu_0\) is the magnetic permeability in a vacuum.

Relate the Fields to the Deflection

The deflection angle \(\theta\) is where the forces from the coil and Earth's magnetic field are balanced. This can be expressed with the tangent function: \[ \text{tan}(\theta) = \frac{B_C}{B_H} \]

Combine Equations

Substituting the equation for \(B_C\) into the tangent relationship gives: \[ \text{tan}(\theta) = \frac{\mu_0 \cdot N \cdot I}{2 \cdot r \cdot B_H} \] This equation describes the deflection angle in terms of the given variables.

Key concepts

Magnetic Fields

A magnetic field is a region where a magnetic force can be detected, typically generated by moving electric charges.
For a tangent galvanometer, this field is created by the flow of current through its wire coil. When current flows through the coil's turns, it produces a magnetic field that interacts with the compass needle nestled in the center of the coil.
Additionally, the Earth's natural magnetic field also plays a role, as it interacts with the coil's magnetic field to affect the compass needle's position.

• Magnetic fields exert forces on magnetic materials and other fields.
• They are essential for understanding the behavior of magnets and for devices like the tangent galvanometer.

Understanding how these fields interact helps explain the principles behind measuring current with this device.

Current Measurement

The tangent galvanometer is an ingenious device for measuring electrical current.
It relies on the way current affects magnetic fields. When you pass current through the coil in the tangent galvanometer, it influences the Earth's magnetic field, deflecting the compass needle.
By measuring this deflection angle, the amount of current can be determined.

• It's a simple yet effective method of measuring electrical current.
• Deflection provides a visual indication of the current's strength.

The degree of deflection is directly proportional to the current, making it a straightforward measurement approach.

Deflection Angle

In the context of the tangent galvanometer, the deflection angle, \(\theta\), is critical for determining the current's strength.
This angle measures how far the compass needle has moved from its original position due to the magnetic field created by the current in the coil.
The formula for the deflection angle relies on the tangent function: \[ \text{tan}(\theta) = \frac{B_C}{B_H} \]The quotient represents the ratio of the coil's magnetic field to Earth's magnetic field.

• The angle \(\theta\) is a key measurement reflecting the balance of forces at play.
• Understanding this angle helps translate deflection into practical current measurements.

A deeper understanding of this concept is vital for effectively using a tangent galvanometer.

Coil Turns

The number of coil turns, represented by \(N\), is a crucial factor in how the tangent galvanometer functions.
More coil turns intensify the magnetic field produced by the current flowing through it.
This field is mathematically given by: \[ B_C = \frac{\mu_0 \cdot N \cdot I}{2 \cdot r} \]

• Each additional turn increases the total magnetic field strength.
• This enhances the device's sensitivity to changes in current.

Understanding how the number of turns affects the field can greatly enhance the precision of current measurements. It's this principle that magnifies the current's magnetic effect, making it more detectable.

Earth's Magnetic Field

Earth's magnetic field is a natural, ever-present force that surrounds our planet.
In the tangent galvanometer, it serves as a balancing force against the field generated by the coil.
Represented as \(B_H\), Earth's magnetic field influences how much the compass needle deflects when current flows.

• It provides a counterbalance to the coil's magnetic field.
• This interaction aids in determining the current's exact strength.

By understanding Earth's magnetic influence, users can effectively use the tangent galvanometer to measure electrical currents with precision. The balance between these fields dictates the deflection angle, key for current measurement.