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Chapter 28: Problem 34

A tangent galvanometer has coil of 50 turns and a mean diameter of \(22 \mathrm{~cm}\). The current through it when the needle is deflected through \(60^{\circ}\) at a place where horizontal components of earth is \(H=30 \mu_{0} \mathrm{~A} / \mathrm{m}\), is : (a) \(300 \mathrm{~mA}\) (b) \(130 \mathrm{~mA}\) (c) \(228 \mathrm{~mA}\) (d) \(158 \mathrm{~mA}\)

Short Answer

Expert verified
The current is 130 mA, corresponding to option (b).

Step by step solution

01

Understand the concept

A tangent galvanometer is a device that measures current strength by using the Earth's magnetic field and a coil. The angle of deflection relates to the magnetic field created by the current in the coil, which contends with the Earth's horizontal magnetic field component. The relationship is given by the tangent galvanometer's formula.
02

Identify given values

The problem gives the following values: - Number of turns in the coil ( ) = 50 - Mean diameter ( ) = 22 cm, so the radius ( ) = 11 cm = 0.11 m - Deflection angle ( ) = 60掳 - Horizontal component of Earth's magnetic field ( ) = 30 渭A/m = 30 脳 10^{-6} A/m.
03

Apply tangent galvanometer formula

The formula for the tangent galvanometer is:\[ I = \frac{{H \cdot an \theta}}{{n \cdot \left(\frac{\mu_0 \cdot N}{2\cdot \pi \cdot r}\right)}}\]where:- \( I \) is the current- \( H \) is the horizontal component of Earth's magnetic field- \( \theta \) is the deflection angle- \( n \) is the number of turns- \( \mu_0 \) is the permeability of free space (- \( N \) is the total turns in the coil- \( r \) is the radius of the coil.
04

Substitute known values into the formula

Substituting the known values into the formula:\[I = \frac{30 \times 10^{-6} \cdot \tan 60掳}{50 \cdot \left(\frac{4\pi \times 10^{-7} \cdot 50}{2\pi \times 0.11}\right)}\]\( \tan 60掳 = \sqrt{3} \). Simplifying gives:\[I = \frac{30 \times 10^{-6} \cdot \sqrt{3}}{50 \cdot \left(\frac{2 \times 50}{0.22}\right) \times 10^{-7}}\]
05

Calculate the current

Perform the calculations:\[I = \frac{30 \times 10^{-6} \cdot 1.732}{50 \cdot \left(\frac{2 \times 50}{0.22}\right) \times 10^{-7}}\]\[I = \frac{51.96 \times 10^{-6}}{\left(\frac{100}{0.22} \right) \times 10^{-7}}\]\[I = \frac{51.96}{454.55} \times 10^{-7}\]\[I \approx 0.0013 \text{ A } = 130 \text{ mA}\].
06

Verify and conclude

After performing the calculations, the closest option to our calculated current is 130 mA. This matches option (b). Conclude that the current when the galvanometer needle is deflected at 60掳 is approximately 130 mA.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Field
A magnetic field is a crucial aspect when discussing a tangent galvanometer. It can be thought of as an invisible field that exerts a force on charges that are moving, like the electrons flowing in an electrical circuit.
The Earth's magnetic field, particularly the horizontal component, is a key player in the operation of a tangent galvanometer.
  • The Earth鈥檚 magnetic field consists of both horizontal and vertical components.
  • The horizontal component, denoted as 'H', is what interacts with the magnetic field generated by the coil.
  • This interaction helps determine the current flowing through the galvanometer by affecting the needle鈥檚 deflection.
Understanding these interactions is crucial for interpreting how measurements with a tangent galvanometer result from the balance between these magnetic forces.
Current Measurement
In the context of a tangent galvanometer, current measurement is about determining the strength of the current passing through its coil. The relationship between the magnetic fields and current is expressed in the tangent galvanometer鈥檚 formula.
Here, the current 'I' is tied to several factors.
  • The horizontal magnetic field affects the deflection.
  • The angle of deflection, the coil turns, and its radius also play significant roles.
  • Formula: \( I = \frac{{H \cdot \tan \theta}}{{n \cdot \left(\frac{\mu_0 \cdot N}{2\cdot \pi \cdot r}\right)}} \)
The formula tells us that the current can be found if we know the deflection and the physical characteristics of the galvanometer. Understanding this formula is pivotal for calculating how much current is being measured.
Coil Turns
The number of turns in a coil is an important parameter in a tangent galvanometer. It refers to how many times the wire is looped around in the coil setup.
This influences the effectiveness of the galvanometer considerably.
  • More turns increase the coil's magnetic field strength.
  • This stronger field intensifies the effect on the needle's deflection.
  • In our example, the coil has 50 turns, impacting the calculation of current.
The number of coil turns, 'N', figures directly into the tangent galvanometer formula, influencing how the electromagnetic forces compete with Earth's magnetic field.
Deflection Angle
In a tangent galvanometer, the deflection angle is the angle which the needle moves from its initial resting position due to the magnetic field produced by the current in the coil. This deflection angle is symbolized by \(\theta\).
This angle is directly related to the strength of the current flowing through the coil.
  • If the current is stronger, the magnetic field will be larger, resulting in a greater deflection angle.
  • In our problem, an angle of \(60^\circ\) was used, which is typical when a substantial current flows.
  • The tangent of this angle, \(\tan 60^\circ = \sqrt{3}\), is used in calculations to determine current strength.
Understanding the concept of the deflection angle is essential for interpreting the results of a tangent galvanometer, as it mirrors the interaction between the electrical and magnetic forces at work.

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