Problem 44 - Master Resource Book in JEE Main Physics

Question

A circular coil of diameter \(21 \mathrm{~cm}\) is held in a magnetic field of induction \(10^{-4} \mathrm{~T}\). The magnitude of magnetic flux linked with the coil when the plane of the coil makes an angle of \(30^{\circ}\) with the field is (a) \(3.1 \times 10^{-6} \mathrm{~Wb}\) (b) \(1.414 \mathrm{~Wb}\) (c) \(1.73 \times 10^{-6} \mathrm{~Wb}\) (d) \(14.14 \mathrm{~Wb}\)

Step-by-Step Solution

Answer

The magnitude of the magnetic flux is \(3.1 \times 10^{-6} \; \text{Wb}\), option (a).

Step 1: Calculate the Radius of the Coil

The diameter of the coil is given as \(21\; \text{cm}\). The radius \(r\) can be calculated by dividing the diameter by 2:\[r = \frac{21}{2} = 10.5\; \text{cm} = 0.105\; \text{m}\]

Step 2: Calculate the Area of the Coil

The area \(A\) of a circle is calculated using the formula \(A = \pi r^2\). Substitute the value of \(r\) from Step 1:\[A = \pi (0.105)^2 \A \approx 3.465 \times 10^{-2} \; \text{m}^2\]

Step 3: Determine the Magnetic Flux Formula

The magnetic flux \(\Phi\) through the coil is given by the formula:\[ \Phi = B \cdot A \cdot \cos(\theta)\]where \(B\) is the magnetic field strength, \(A\) is the area, and \(\theta\) is the angle between the magnetic field and the normal to the plane of the coil.

Step 4: Plug in the Values

Substitute \(B = 10^{-4} \; \text{T}\), \(A \approx 3.465 \times 10^{-2} \; \text{m}^2\), and \(\theta = 30^\circ\) into the magnetic flux formula:\[\Phi = 10^{-4} \cdot 3.465 \times 10^{-2} \cdot \cos(30^\circ) \]

Step 5: Calculate \(\cos(30^\circ)\)

The value of \(\cos(30^\circ)\) is \(\frac{\sqrt{3}}{2}\), approximately \(0.866\).

Step 6: Calculate the Magnetic Flux

Substitute \(\cos(30^\circ) \approx 0.866\) into the equation from Step 4?\[\Phi = 10^{-4} \cdot 3.465 \times 10^{-2} \cdot 0.866 \\Phi \approx 3.1 \times 10^{-6} \; \text{Wb}\]

Step 7: Determine the Correct Answer

The magnitude of the magnetic flux linked with the coil is \(3.1 \times 10^{-6} \; \text{Wb}\), which corresponds to option (a).

Key Concepts

Circular Coil

A circular coil is a loop or ring shaped in a circle, which can be composed of wire or other conducting material. These coils are often used in applications involving electromagnetism, such as inductors or magnets. In our exercise, we have a coil with a diameter of 21 cm. To use this coil to calculate magnetic flux, we first find its radius, which is half of the diameter. The radius is 10.5 cm or 0.105 meters when expressed in meters.
The reason the circular coil is significant in this context is that it determines the area through which the magnetic field lines pass. The area of this coil will directly influence how much magnetic flux is generated, as seen in the formula for magnetic flux, which involves area as a component. Knowing how to calculate the area of a circular coil can help you determine its interaction with a magnetic field.

Magnetic Field

The magnetic field is an invisible field around a magnet where magnetic forces are exerted. It is represented by the letter "B" in scientific equations and measured in teslas (T). In our example, the magnetic field strength is given as \(10^{-4} \,\text{T}\).
This field represents the density of magnetic lines of force at a given point. The stronger the magnetic field, the more force it can exert on magnetic materials or coils. In terms of our exercise, the magnetic field strength directly impacts the magnetic flux. The stronger the field, with other factors being constant, the larger the flux. Understanding how magnetic fields operate is pivotal in assessing how much energy they can transfer through a given area.

Angle of Inclination

The angle of inclination refers to the angle formed between the plane of the coil and the direction of the magnetic field. In our example, this angle is given as \(30^{\circ}\). The angle of inclination is crucial because it affects how effectively the magnetic field passes through the coil.
The concept to grasp here is that the effective component of the magnetic field interacting with the plane of the coil is reduced by the cosine of this angle. This means not all the field passes through the coil perpendicularly; only a portion, dictated by the angle, contributes to the magnetic flux. The closer the angle of inclination is to zero, the more aligned the field is with the normal to the plane of the coil, and the larger the resultant magnetic flux. This angle is a key factor in calculating the actual magnetic flux linked with the coil.

Cosine Function

The cosine function, written as \(\cos(\theta)\), is a trigonometric function that is crucial for determining how the angle of inclination reduces the contribution of the magnetic field in the flux calculation. In the given problem, \(\theta\) equals \(30^{\circ}\), and \(\cos(30^{\circ})\) is approximately 0.866.
The reason the cosine function is used is that it provides a way to calculate the effective strength of the magnetic field that passes through the coil based on the angle of inclination. When the angle is zero, \(\cos(0) = 1\), indicating that the entire magnetic field contributes to the flux. As the angle increases, while the other parameters remain constant, the cosine value decreases, reducing the effective magnetic field and consequently the flux. Understanding this function's role is central to calculating magnetic flux accurately in such contexts.