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Chapter 16: Problem 43

When the number of turns in a coil is doubled without any change in the length of the coil, its self-inductance becomes (A) Four times (B) Doubled (C) Halved (D) Squared

Short Answer

Expert verified
The self-inductance of a coil becomes four times its initial value when the number of turns is doubled without any change in the length of the coil. The correct answer is (A) Four times.

Step by step solution

01

Self-Inductance Formula

The formula for self-inductance of a coil is given by: \(L = \mu_0 \frac{N^2 A}{l}\), where \(L\) is the self-inductance, \(\mu_0\) is the permeability of free space, \(N\) is the number of turns, \(A\) is the cross-sectional area, and \(l\) is the length of the coil. #Step 2: Determine self-inductance when the number of turns is doubled#
02

New Self-Inductance with Doubled Number of Turns

Now let's determine the new self-inductance when we double the number of turns, which means \(N \) will be replaced with \(2N\): \(L' = \mu_0 \frac{(2N)^2 A}{l}\) #Step 3: Compare the new self-inductance with the original self-inductance#
03

Comparing Self-Inductance Values

Divide the new self-inductance (\(L'\)) by the original self-inductance (\(L\)): \(\frac{L'}{L} = \frac{\mu_0 \frac{(2N)^2 A}{l}}{\mu_0 \frac{N^2 A}{l}}\) #Step 4: Simplify the ratio to find the change in self-inductance#
04

Simplify the Ratio

After canceling out the common terms and simplifying, \(\frac{L'}{L} = \frac{(2N)^2}{N^2} = \frac{4N^2}{N^2} = 4\) The self-induction becomes four times its initial value when the number of turns is doubled, without any change in the length of the coil. So, the correct answer is: (A) Four times

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Key Concepts

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

Coil
A coil is a fundamental component in electromagnetism, consisting of wire wound into a spiral or helix. Coils have several practical applications in electronics, such as in transformers and inductors. The structure of a coil allows it to create a magnetic field when electrified.
This magnetic field can induce electrical currents in nearby conductors, hence it is central to many electrical devices.
Important characteristics of a coil include:
  • Resistance
  • Inductance
  • Number of turns
  • Coil length
Each of these factors can affect how the coil behaves in an electrical circuit. Coils are versatile, used extensively in electromagnets and have roles in shaping magnetic fields efficiently.
Number of Turns
The number of turns in a coil is crucial as it influences the coil's inductance. Inductance is a measure of the coil's ability to resist changes in electric current and is impacted significantly by the number of turns.
When the number of turns is increased, the strength of the magnetic field produced by the coil also increases.
The formula for self-inductance is given as:\(L = \mu_0 \frac{N^2 A}{l}\),where:
  • \(L\) is the self-inductance
  • \(\mu_0\) is the permeability of free space
  • \(N\) is the number of turns
  • \(A\) is the cross-sectional area
  • \(l\) is the length of the coil
The formula shows that the inductance is proportional to the square of the number of turns. Doubling the number of turns increases the inductance by a factor of four (since \( (2N)^2 = 4N^2 \)), illustrating the importance of turns in determining a coil's behavior in an electrical circuit.
Electromagnetism
Electromagnetism is a branch of physics that explores the interaction between electric fields and magnetic fields. A coil becomes an effective tool in these interactions due to its ability to generate magnetic fields when electric current flows through it.
Some key principles include:
  • Faraday's Law of Induction
  • Self-inductance
  • Magnetic field creation
Using a coil, electromagnetism can be harnessed to perform tasks like generating electricity, transmitting signals, and powering motors. The relationship between electricity and magnetism forms the foundation for much of the modern technology we rely on today. The interplay of changing electric and magnetic fields underlies the operation of transformers, inductors, and many other devices, making electromagnetism a vital study area for engineers and physicists alike.

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Most popular questions from this chapter

A conducting rod of length \(2 \ell\) is rotating with constant angular speed \(\omega\) about its perpendicular bisector. A uniform magnetic field \(\vec{B}\) exists parallel to the axis of rotation. The EMF induced between two ends of the rod is (A) \(B \omega \ell^{2}\) (B) \(\frac{1}{2} B \omega \ell^{2}\) (C) \(\frac{1}{8} B \omega \ell^{2}\) (D) Zero

A \(50 \mathrm{mH}\) coil carries a current of \(2 \mathrm{~A}\), the energy stored in it in \(\mathrm{J}\) is (A) \(0.05\) (B) \(0.1\) (C) \(0.5\) (D) 1

A current \(i\) ampere flows along an infinitely long straight thin-walled tube, then the magnetic induction at any point inside the tube at a distance \(r\) from centre is (A) Infinite (B) Zero (C) \(\frac{\mu_{0}}{4 \pi} \cdot \frac{2 i}{r}\) (D) \(\frac{2 i}{r}\)

Two coaxial solenoids are made by winding thin insulated wire over a pipe of cross-sectional area \(A=10 \mathrm{~cm}^{2}\) and length \(=20 \mathrm{~cm}\). If one of the solenoid has 300 turns and the other 400 turns, their mutual inductance is \(\left(\mu_{0}=4 \pi \times 10^{-7} \mathrm{TmA}^{-1}\right)\) (A) \(2.4 \pi \times 10^{-5} \mathrm{H}\) (B) \(4.8 \pi \times 10^{-4} \mathrm{H}\) (C) \(4.8 \pi \times 10^{-5} \mathrm{H}\) (D) \(2.4 \pi \times 10^{-4} \mathrm{H}\)

A coil is suspended in a uniform magnetic field, with the plane of the coil parallel to the magnetic lines of force. When a current is passed through the coil, it starts oscillating; it is very difficult to stop. But if an aluminium plate is placed near to the coil, it stops. This is due to (A) development of air current when the plate is placed. (B) induction of electrical charge on the plate. (C) shielding of magnetic lines of force as aluminium is a paramagnetic material. (D) electromagnetic induction in the aluminium plate giving rise to electromagnetic damping.
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