Question

A long wire carr1es a steady current. It is bent into a circle of one turn and the magnetic field at the centre of the coil is \(\mathrm{B}\). It is then bent into a circular Loop of n turns. The magnetic field at the centre of the coil for same current will be. (a) \(\mathrm{nB}\) (b) \(\mathrm{n}^{2} \mathrm{~B}\) (c) \(2 \mathrm{nB}\) (d) \(2 \mathrm{n}^{2} \mathrm{~B}\)

Short answer

The magnetic field at the center of the coil for the same current with 'n' turns will be (a) nB.

Step-by-step solution

Recall relationship between magnetic field and coil attributes

The magnetic field at the center of a circular loop is given by the formula: \[B = \frac{\mu_0 * I * R}{2 R^2}\] Where: - \(B\) is the magnetic field at the center of the loop, - \(\mu_0\) is the permeability of free space, - \(I\) is the current through the wire, - \(R\) is the radius of the circular loop. In this case, when the coil has one turn, the magnetic field is given as \(B\).

Find the expression for magnetic field with n turns

Now we need to extend this expression for a coil with 'n' turns. The magnetic field produced by each turn is the same, given by the formula in step 1. Since the loops are wound on top of each other, the magnetic field at the center will add up. Thus, the total magnetic field at the center of the coil with 'n' turns would be: \[B_{total} = n * B\] Where: - \(B_{total}\) is the total magnetic field at the center, - \(n\) is the number of turns. Now let's find the correct answer from the given options.

Match the result with given options

Comparing our derived expression with the given options, we can see that: \[B_{total} = n * B\] Corresponds to option (a) nB. So, the magnetic field at the center of the coil for the same current with 'n' turns will be: (a) nB

Key concepts

Ampere's Law

Ampere's Law is a fundamental principle used to understand the behavior of magnetic fields in various conditions. It relates the integrated magnetic field around a closed loop to the current passing through the loop. Mathematically, it is expressed as:\[\oint \vec{B} \cdot d\vec{l} = \mu_0 I\]where:

• \(\vec{B}\) is the magnetic field.
• \(d\vec{l}\) is the differential length along the loop.
• \(\mu_0\) is the permeability of free space.
Ampere's Law helps in simplifying complex magnetic field calculations by focusing only on current passing through an enclosed region. It is particularly useful in situations with high symmetry such as a solenoid—a coil wound into a tightly packed helix. In cases like these, Ampere's Law can predict how magnetic fields behave and are distributed within and around the coil.

Permeability of Free Space

One of the constant figures in magnetic field calculations is the permeability of free space, denoted as \(\mu_0\). This value is critical as it determines how much a magnetic field can penetrate and exist in a vacuum. The permeability of free space has a precise value of \(4\pi \times 10^{-7}\, \text{T}\cdot\text{m/A}\), and it serves as a measure of the magnetic response of a vacuum to a magnetic field. This constant is used across various electromagnetic equations, including those utilized in Ampere's Law, Maxwell's Equations, and more.The consistency of \(\mu_0\) means that it's a universal standard in physics, making it central to understanding and calculating forces in electromagnetics, especially those involving steady currents and magnetostatics.

Current in a Wire

Current flowing through a wire is one of the fundamental principles when discussing magnetic fields. The actual movement of electric charge, or current (\(I\)), influences the generation of a magnetic field. According to the right-hand rule, if you curl the fingers of your right hand in the direction of the current, your thumb will point in the direction of the magnetic field lines.The magnitude and direction of the magnetic field created by a current-carrying wire depend on multiple factors:

• The amount of current flowing through the wire; more current results in a stronger magnetic field.
• The distance from the wire; the field strength decreases with increasing distance from the wire.
• The geometry of the wire, since wires shaped into loops or coils can enhance and change the pattern of the magnetic field produced.

Understanding the relationship between current and magnetic fields is essential for analyzing scenarios such as those presented in the exercise with circular loops, making it a cornerstone of electromagnetic studies.

Magnetic Field and Coil Attributes

The attributes of a coil, like the number of turns, the radius, and the current, significantly affect the magnetic field strength at its center. In circular loops, the magnetic field intensity is directly proportional to the number of turns and the current passing through the wire.### Calculating the Magnetic FieldThe formula to calculate the magnetic field \(B\) at the center of a loop that has a single turn, considering it has current \(I\) and radius \(R\), is given by:\[B = \frac{\mu_0 I}{2R}\]For coils with \(n\) turns, all similar loops contribute equally to the total magnetic field. Hence, the field becomes:\[B_{total} = nB = \frac{\mu_0 n I}{2R}\]The factor of \(n\) arises from the additive effect of each loop's magnetic field. ### Practical ImplicationsIncreasing the number of turns increases the resultant magnetic field, which has practical applications in electromagnets, transformers, and inductors. By understanding how these attributes affect the field, we can design devices to have desired magnetic properties, essential for modern technology.