Problem 1985 - JEE Physics

Question

When 2 Amp current is passed through a tangent galvanometer. It gives a deflection of \(30^{\circ} .\) For \(60^{\circ}\) deflection, the current must be (a) \(1 \mathrm{Amp}\) (b) \(2 \sqrt{3 a m p}\) (c) \(4 \mathrm{amp}\) (d) \(6 \mathrm{Amp}\)

Step-by-Step Solution

Answer

The current required for a \(60^{\circ}\) deflection in a tangent galvanometer is \(2\sqrt{3}Amp\).

Step 1: Understand the tangent galvanometer

A tangent galvanometer is an instrument used to measure the current by the deflection of a magnetic needle. The current flowing through the coil of the galvanometer creates a magnetic field, causing the needle to deflect. The deflection angle is proportional to the strength of the current.

Step 2: Derive the relationship between current and deflection angle

In a tangent galvanometer, the tangent of the deflection angle is directly proportional to the current. This can be represented as: \( \tan\theta=kI \) where: θ = deflection angle I = current k = a constant For our given problem, we have a deflection angle of 30° corresponding to a current of 2 Amps.

Step 3: Determine the constant k

To find the value of k, we can use the initial data provided in the problem: \( \tan30^{\circ}= k(2) \) \( \frac{1}{\sqrt{3}}=2k \) Solve for k: \( k =\frac {1}{2\sqrt{3}} \)

Step 4: Calculate the current for the 60° deflection

Now we can use the equation we derived earlier to find the current corresponding to a 60° deflection: \( \tan60^{\circ}= kI \) Use the value of k we found previously: \( \sqrt{3} = \frac{1}{2\sqrt{3}}I \) Solve for I: \( I = 2\sqrt{3}Amps \)

Step 5: Choose the correct answer

The current for a 60° deflection is 2√3 Amp. The correct answer is: (b) \(2\sqrt{3}Amp\)

Key Concepts

Current Measurement

Measuring current accurately is crucial in many electrical and electronic applications. One fascinating tool for current measurement is the tangent galvanometer. This device allows us to determine the electric current passing through a coil by observing the behavior of a magnetic needle. When current flows through the coil of the tangent galvanometer, it produces a magnetic field. This field interacts with the earth's magnetic field, causing a small needle inside the device to deflect. By measuring this deflection, we can infer the current through the coil. This method is unique because it doesn't rely on the typical circuit elements like resistors or capacitors for measurement. Instead, it uses the physical principles of magnetism.

Deflection Angle

The deflection angle in a tangent galvanometer is the angle by which the magnetic needle rotates due to the influence of the magnetic field created by the current. This angle provides us with vital information about the strength of the current. In the exercise we have, for example, a deflection angle of 30° corresponds to a 2 Amp current. The relationship is not just random; it's based on a clear mathematical relationship. This angle helps us to calculate the current using known values and a derived constant. Understanding the deflection angle is key to harnessing the full potential of a tangent galvanometer in current measurement applications.

Magnetic Field

The concept of the magnetic field is central to understanding the workings of the tangent galvanometer. When current flows through the coil, it generates a magnetic field, influencing the magnetic needle inside the device. This magnetic field is proportional to the amount of current flowing through the coil. It's important to note that the magnetic field doesn't just exist in isolation. It interacts with external magnetic fields, such as the Earth's, which affects how the needle deflects. By controlling and measuring these fields, the tangent galvanometer can give us precise readings about the current flowing through the circuit. This illustrates the power and versatility of electromagnetic principles in practical measuring instruments.

Proportional Relationship

A crucial principle in the function of tangent galvanometers is the proportional relationship between the current and the tangent of the deflection angle. Mathematically, this is depicted as \( \tan\theta=kI \), where \( \theta \) is the deflection angle, \( I \) is the current, and \( k \) is a constant specific to the device's configuration. Essentially, if you know \( k \) and the deflection angle, you can compute the unknown current. This proportionality allows us to solve problems like the one in the exercise, where changing the deflection angle results in a different current value. The core idea here is the direct relationship that exists, making the tangent galvanometer an elegant and effective tool for measuring currents.