Two coplanar and concentric circular loops are of radii 0·5 cm and 11 cm, respectively. These loops are placed in a uniform external magnetic field of 0·4 T acting perpendicular to their plane. Calculate the mutual inductance of the arrangement .
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NCERT physics class 12
Answers
Answer: M = 9.3 * 10^(-8) H
Explanation:
Given: R1 = 0.5cm and R2 = 11cm
External magnetic Field, B = 0.4T
Mutual Inductance between two coils having radii of R1 and R2 is given by:
M = μ * pi * (R1 * R2) / (R1 + R2)
where:
M is the mutual inductance
μ is the magnetic permeability of free space (approximately 4π * 10^(-7) Tm/A)
R1 and R2 are the radii of the smaller and larger loop, respectively
In this case, R1 = 0.5 cm = 0.005 m and R2 = 11 cm = 0.11 m
Substituting the values in the formula:
M = 4 * pi * 10^(-7) * pi * (0.005 * 0.11) / (0.005 + 0.11)
M = 9.3 * 10^(-8) H
Hence, the required value of mututal inductance of the arrangement is 9.3*10^-8 H.
Mutual induction is defined as the property of the coils that enables it to oppose the changes in the current in another coil. With a change in the current of one coil, the flow changes too thus inducing EMF in the other coil. This phenomenon is known as mutual induction.
With a smooth current in one coil, as given in the diagram, a magnetic field is formed in the other coil. That magnetic field remains unchanged and hence, according to Faraday’s law, it will not lead to any voltage creation in the secondary coil. Now, if we open the switch so as to stop the current, the magnetic field will change in the coil placed on the right-hand side, thus inducing a voltage. The coil does not support changes and hence the induced voltage will promote a flow of current in the secondary coil trying to maintain the already existing magnetic field. The fact, however, remains that the induced field will always repel the change.
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Answer: The mutual inductance of the arrangement is:
M = μ0 * N^2 * A1 / l ≈ 2.37 × 10^-8 H
Explanation: The mutual inductance between two concentric circular loops can be calculated using the following formula:
- M = μ0 * N1 * N2 * A / l
where:
μ0 is the permeability of free space, which is equal to 4π × 10^-7 H/m
N1 and N2 are the number of turns in the two loops
A is the area of one of the loops
l is the distance between the two loops
In this problem, the two loops are concentric and coplanar, so their distance l is equal to the difference between their radii:
- l = R2 - R1 = 11 cm - 0.5 cm = 10.5 cm = 0.105 m
The area of each loop is given by:
- A = π * r^2
where r is the radius of the loop. Therefore:
- A1 = π * (0.005 m)^2 = 7.85 × 10^-5 m^2 A2 = π * (0.11 m)^2 = 3.8013 × 10^-2 m^2
The number of turns in each loop is not given in the problem statement. Let's assume that each loop has N turns.
Finally, we can plug all these values into the formula for mutual inductance:
- M = μ0 * N^2 * A1 / l = μ0 * N^2 * A2 / l
Setting these two expressions for M equal to each other, we can solve for N:
- N = √(μ0 * A1 * A2 / l^2)
Plugging in the values, we get:
- N = √(4π × 10^-7 H/m * 7.85 × 10^-5 m^2 * 3.8013 × 10^-2 m^2 / (0.105 m)^2)
- N ≈ 1.48
Therefore, the mutual inductance of the arrangement is:
- M = μ0 * N^2 * A1 / l ≈ 2.37 × 10^-8 H
Learn more about mutual inductance here - https://brainly.in/question/925501
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