Problem 12 - Master Resource Book in JEE Main Physics

Question

A circular coil \(A\) of radius \(r\) carries current \(i\). Another circular coil \(B\) of radius \(2 r\) carries current of \(i\). The magnetic fields at the centres of the circular coils are in the ratio of (a) \(3: 1\) (b) \(4: 1\) (c) \(1: 1\) (d) \(2: 1\)

Step-by-Step Solution

Answer

The magnetic fields are in the ratio of 2:1, option (d).

Step 1: Understanding the Formula for Magnetic Field at the Center of a Circular Coil

The magnetic field at the center of a circular coil carrying current is given by the formula \( B = \frac{\mu_0 i}{2r} \), where \( \mu_0 \) is the permeability of free space, \( i \) is the current, and \( r \) is the radius of the coil.

Step 2: Applying the Formula to Coil A

For coil \( A \), which has a radius \( r \) and carries current \( i \), the magnetic field at its center is \( B_A = \frac{\mu_0 i}{2r} \).

Step 3: Applying the Formula to Coil B

For coil \( B \), which has a radius of \( 2r \) and carries the same current \( i \), the magnetic field at its center is \( B_B = \frac{\mu_0 i}{2(2r)} = \frac{\mu_0 i}{4r} \).

Step 4: Calculating the Ratio of Magnetic Fields

Now, find the ratio of magnetic fields at the centers of the two coils: \( \frac{B_A}{B_B} = \frac{\frac{\mu_0 i}{2r}}{\frac{\mu_0 i}{4r}} \). This simplifies to \( \frac{4r}{2r} = 2 \), so the ratio is \( 2:1 \).

Key Concepts

Circular Coil

A circular coil is a loop of wire, typically shaped in a circle, that carries an electric current. When an electric current flows through the coil, it produces a magnetic field. This is a common occurrence in many electrical devices, such as motors and transformers. The circular shape allows the current to create a uniform magnetic field around the center of the coil.

Understanding the behavior of this magnetic field is crucial in physics and engineering. The characteristics of the magnetic field depend on various factors, including the number of turns in the coil, the current flowing through the coil, and the radius of the coil.

• The number of turns: More loops enhance the magnetic field strength.
• Current: Increasing the current also increases the magnetic field.
• Radius: The relation of the radius to the achieved magnetic field is inversely proportional, as explained later.

A circular coil's ability to produce a magnetic field makes it an integral part of numerous applications, from electromagnets to induction cooktops.

Magnetic Field Formula

The magnetic field's strength at the center of a circular coil is a key concept defined by the formula: \[ B = \frac{\mu_0 \cdot i}{2r} \] Here,

• \( B \) is the magnetic field strength,
• \( \mu_0 \) represents the permeability of free space,
• \( i \) is the current flowing through the coil,
• \( r \) denotes the radius of the coil.

The permeability of free space, \( \mu_0 \), is a constant value that helps in determining how easily a magnetic field can pass through a void or vacuum. This formula shows that the magnetic field strength at the center depends directly on the current and inversely on the radius of the coil. Understanding how to apply and manipulate this formula is essential for solving problems related to magnetic fields in coils.

Current and Magnetic Field Relationship

The relationship between the current flowing through a coil and the resulting magnetic field is fundamental in electromagnetism. When electric current passes through a coil, it generates a magnetic field perpendicular to the flow of current. This magnetic field's strength is directly proportional to the magnitude of the current. This means that:

• Increasing the current will strengthen the magnetic field.
• Decreasing the current will weaken the magnetic field.

Additionally, if the direction of the current changes, the direction of the produced magnetic field will also reverse. This direct relationship is at the heart of technologies like electromagnets, where the magnetic field can be controlled and manipulated simply by varying the current. It is this principle that allows for the creation of diverse applications, such as MRI machines, which harness magnetic fields generated by coils to image the human body.

Radius and Magnetic Field

The radius of a circular coil affects the magnetic field strength at its center. In the magnetic field formula \( B = \frac{\mu_0 i}{2r} \), it is evident that the magnetic field \( B \) is inversely proportional to the radius \( r \). This means:

• A smaller radius results in a stronger magnetic field, as the field lines are closer together at the center.
• A larger radius leads to a weaker magnetic field because the field spreads out more over a larger area.

For instance, if you double the radius of a coil while keeping the current constant, the magnetic field's strength at the center is halved. This relationship is pivotal when designing electromagnets and electrical circuits to ensure the optimal magnetic field strength is achieved for the intended application. Understanding this concept allows engineers and physicists to predict and adjust the behavior of devices based on the geometric configuration of the coils involved.