Reduction Factor of Tangent Galvanometer: Definition, Formula, Derivation and Unit

The reduction factor of a tangent galvanometer is an important parameter used to determine the electric current flowing through its circular coil. It establishes the relationship between the current passing through the coil and the tangent of the angular deflection produced by the magnetic needle.

A tangent galvanometer works on the tangent law of magnetism. When electric current flows through its circular coil, a magnetic field is generated at the centre. This field acts perpendicular to the horizontal component of Earth’s magnetic field, causing the compass needle to deflect.

The reduction factor helps calculate the current corresponding to a measured deflection angle. It depends on the radius and number of turns of the coil, along with the horizontal component of Earth’s magnetic field.

What Is the Reduction Factor of a Tangent Galvanometer?

The reduction factor of a tangent galvanometer is defined as the current required to produce a tangent of the deflection angle equal to unity.

In other words, it is the current required to produce a deflection of 45° in a properly aligned tangent galvanometer.

The mathematical expression is:

K=Itan⁡θ\boxed{K=\frac{I}{\tan\theta}}

Where:

  • K = Reduction factor of the tangent galvanometer
  • I = Current flowing through the coil (A)
  • θ = Angular deflection of the magnetic needle

When the deflection angle is 45°:

tan⁡45∘=1\tan45^\circ=1

Therefore,

K=I\boxed{K=I}

Thus, the reduction factor is numerically equal to the current required to produce a deflection of 45°.

Formula for Reduction Factor of Tangent Galvanometer

The reduction factor can be calculated using the following formula:

K=2rBHμ0N\boxed{K=\frac{2rB_H}{\mu_0N}}

Where:

  • K = Reduction factor (A)
  • r = Mean radius of the circular coil (m)
  • BHB_H = Horizontal component of Earth’s magnetic field (T)
  • μ₀ = Permeability of free space (4π×10−7H/m)(4\pi\times10^{-7} H/m)
  • N = Number of turns in the circular coil

The formula shows that the reduction factor depends on the physical dimensions of the coil and the strength of Earth’s horizontal magnetic field.

Derivation of Reduction Factor of Tangent Galvanometer

To derive the reduction factor formula, consider a tangent galvanometer consisting of a circular coil having N turns and a mean radius r. Let a current I flow through the coil.

Step 1: Magnetic Field at the Centre of the Coil

According to the Biot-Savart law, the magnetic field produced at the centre of a circular coil carrying current is:

B=μ0NI2rB=\frac{\mu_0NI}{2r}

Where:

  • B = Magnetic field at the centre of the coil
  • μ₀ = Permeability of free space
  • N = Number of turns
  • I = Current flowing through the coil
  • r = Radius of the coil

Step 2: Apply the Tangent Law

The plane of the circular coil is adjusted along the magnetic meridian. Therefore, the magnetic field produced by the coil is perpendicular to Earth’s horizontal magnetic field.

According to the tangent law:

tan⁡θ=BBH\tan\theta=\frac{B}{B_H}

Substituting the magnetic field produced by the coil:

tan⁡θ=μ0NI2rBH\tan\theta= \frac{\mu_0NI}{2rB_H}

Rearranging the equation:

I=2rBHμ0Ntan⁡θI=\frac{2rB_H}{\mu_0N}\tan\theta

Step 3: Compare with the Standard Equation

The current through a tangent galvanometer is also expressed as:

I=Ktan⁡θI=K\tan\theta

Comparing both equations:

Ktan⁡θ=2rBHμ0Ntan⁡θK\tan\theta= \frac{2rB_H}{\mu_0N}\tan\theta

Therefore, the reduction factor is:

K=2rBHμ0N\boxed{K=\frac{2rB_H}{\mu_0N}}

This is the required expression for the reduction factor of a tangent galvanometer.

Why Is the Reduction Factor Defined at 45°?

The reduction factor is the current required to produce a tangent of deflection equal to one.

From its definition:

K=Itan⁡θK=\frac{I}{\tan\theta}

For a deflection angle of 45°:

tan⁡45∘=1\tan45^\circ=1

Substituting:

K=I1K=\frac{I}{1}

Therefore:

K=I\boxed{K=I}

Hence, the reduction factor of a tangent galvanometer is the current required to produce a deflection of 45°.

Important: A deflection of 45° is the condition for defining the reduction factor, not the only angle at which a tangent galvanometer can measure current.

Unit of Reduction Factor

The unit of the reduction factor is ampere (A).

This is because:

K=Itan⁡θK=\frac{I}{\tan\theta}

Current is measured in amperes, whereas the tangent of an angle is dimensionless.

Therefore:

Unit of K = Ampere (A)\boxed{\text{Unit of K = Ampere (A)}}

Factors Affecting the Reduction Factor

The reduction factor depends on the construction and operating location of the tangent galvanometer.

1. Radius of the Circular Coil

The reduction factor is directly proportional to the radius of the coil.

K∝rK\propto r

An increase in coil radius increases the reduction factor when all other parameters remain constant.

2. Number of Turns

The reduction factor is inversely proportional to the number of turns.

K∝1NK\propto\frac{1}{N}

Increasing the number of turns reduces the current required to produce the same deflection.

3. Horizontal Component of Earth’s Magnetic Field

The reduction factor is directly proportional to the horizontal component of Earth’s magnetic field.

K∝BHK\propto B_H

Therefore, its value can vary with geographical location.

4. Magnetic Permeability

The reduction factor is inversely proportional to the permeability of the medium represented by μ₀ in the standard formula.

For an air-core tangent galvanometer, μ₀ is taken as the permeability of free space.

Numerical Example on Reduction Factor

Question: A tangent galvanometer has a circular coil of mean radius 0.15 m and 20 turns. If the horizontal component of Earth’s magnetic field is (3.6\times10^{-5}) T, calculate its reduction factor.

Given:

  • Mean radius, r = 0.15 m
  • Number of turns, N = 20
  • Horizontal magnetic field, (BH=3.6×10−5)T(B_H=3.6\times10^{-5}) T
  • Permeability, (μ0=4π×10−7)H/m(\mu_0=4\pi\times10^{-7}) H/m

Formula:

K=2rBHμ0NK=\frac{2rB_H}{\mu_0N}

Substituting the values:

K=2×0.15×3.6×10−54π×10−7×20K=\frac{2\times0.15\times3.6\times10^{-5}} {4\pi\times10^{-7}\times20}
K=1.08×10−58π×10−6K=\frac{1.08\times10^{-5}} {8\pi\times10^{-6}}
K=0.430 A\boxed{K=0.430\ A}

The reduction factor of the tangent galvanometer is approximately 0.430 A. This means that a current of approximately 0.430 A is required to produce a deflection of 45°.

Difference Between Reduction Factor and Sensitivity of Tangent Galvanometer

The reduction factor and sensitivity describe different characteristics of a tangent galvanometer.

Basis Reduction Factor Sensitivity
Meaning Current required for unit tangent of deflection Change in deflection for a change in current
Formula K = I/tan θ dθ/dI
Unit Ampere (A) Radian per ampere (rad/A)
Significance Indicates the current required for a specified tangent value Indicates the instrument’s response to current changes
Lower value Smaller current is needed for unit tangent Greater angular sensitivity for the same current change

Applications of Reduction Factor

The reduction factor is useful in several laboratory applications:

  1. Current measurement: It helps determine unknown current using the measured deflection of a tangent galvanometer.
  2. Instrument calibration: It can be used to establish the relationship between current and angular deflection.
  3. Magnetic field experiments: It helps calculate the magnetic field produced by a current-carrying circular coil.
  4. Determination of Earth’s magnetic field: It is used in experiments involving the horizontal component of Earth’s magnetic field.
  5. Comparison of galvanometers: It helps compare the current requirements of different tangent galvanometers.

Conclusion

The reduction factor of a tangent galvanometer represents the current required to produce a tangent of deflection equal to unity, corresponding to an angular deflection of 45°. It is calculated using the expression (K=2rBH/(μ0N)K=2rB_H/(\mu_0N), where the coil radius, number of turns, and horizontal component of Earth’s magnetic field determine its value. The reduction factor is expressed in amperes and is useful for calculating unknown currents from the measured deflection of a tangent galvanometer.

Read Next:

  1. Conversion of Galvanometer into Voltmeter: Formula, Derivation and Working
  2. Conversion of Galvanometer into Ammeter – Formula, Circuit Diagram and Derivation
  3. Applications of Galvanometer
  4. Galvanometer: Definition, Working Principle, Construction and Working
  5. Sensitivity of Galvanometer: Formula, Types, Factors and Explanation
  6. Figure of Merit of a Galvanometer: Formula, Experiment and Calculation
  7. Galvanometer Resistance: Formula, Calculation and Measurement
  8. Moving Coil Galvanometer: Principle, Construction, Working, Diagram, Formula Derivation
  9. Galvanometer Constant: Formula, Derivation, Definition and Significance
  10. Galvanometer and Direction of Current
  11. Galvanometer Half Deflection Method: Formula, Experiment and Calculation

Leave a comment