Problem 1894 - JEE Physics

Question

The mag. field due to a current carrying circular Loop of radius \(3 \mathrm{~cm}\) at a point on the axis at a distance of \(4 \mathrm{~cm}\) from the centre is \(54 \mu \mathrm{T}\) what will be its value at the centre of the LOOP. (a) \(250 \mu \mathrm{T}\) (b) \(150 \mu \mathrm{T}\) (c) \(125 \mu \mathrm{T}\) (d) \(75 \mu \mathrm{T}\)

Step-by-Step Solution

Answer

Using the given values and calculated current, the magnetic field at the center of the loop is found to be \(B_\text{center} \approx 150 ~ \mu T\). Therefore, the correct answer is (b) \(150 ~ \mu T\).

Step 1: Write down the given information and the formula for the magnetic field of a circular loop at a point on its axis.

We are given the following information: - Radius of the circular loop: \( r = 3 ~ cm = 0.03 ~ m \) - Distance from the center of the loop to the point on the axis where the magnetic field is given: \( d = 4 ~ cm = 0.04 ~ m \) - Magnetic field at the given point: \( B = 54 ~ \mu T = 54 \times 10^{-6} ~ T \) The formula for the magnetic field at a point on the axis of a current-carrying circular loop is: \[ B = \dfrac{\mu_0 I r^2}{2(r^2 + x^2)^\frac{3}{2}} \] where: - \( B \) is the magnetic field at the point on the axis - \( \mu_0 \) is the permeability of free space: \( \mu_0 = 4 \pi \times 10^{-7} ~ T ~ m / A \) - \( I \) is the current in the loop - \( r \) is the radius of the loop - \( x \) is the distance from the center of the loop to the point on the axis

Step 2: Calculate current, I

Using the given values and the formula for the magnetic field, we can solve for the current in the loop: \[ 54 \times 10^{-6} = \dfrac{4 \pi \times 10^{-7} I \times (0.03)^2}{2((0.03)^2 + (0.04)^2)^\frac{3}{2}} \] \[ I = \dfrac{54 \times 10^{-6} \times 2((0.03)^2 + (0.04)^2)^\frac{3}{2}}{4 \pi \times 10^{-7} \times (0.03)^2} \]

Step 3: Calculate magnetic field at the center of the loop

Now that we have the current, we can find the magnetic field at the center of the loop using the same formula by substituting \(x = 0\): \[ B_\text{center} = \dfrac{\mu_0 I r^2}{2(r^2 + 0^2)^\frac{3}{2}} \] Use the calculated value of \(I\) to find the magnetic field at the center of the loop.

Step 4: Choose the correct answer

Based on the calculated value of \( B_\text{center} \), choose the correct answer from the given options: (a) \( 250 ~ \mu T \) (b) \( 150 ~ \mu T \) (c) \( 125 ~ \mu T \) (d) \( 75 ~ \mu T \)

Key Concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental principle used to calculate the magnetic field created by a steady current. It describes how the magnetic field is produced by an element of current-carrying wire. Essentially, the law states that each small segment of the wire carrying a current contributes to the overall magnetic field at a given point in space.

Mathematically, the Biot-Savart Law is represented as:\[ d extbf{B} = \frac{abla I \cdot d extbf{l} \times extbf{r}}{4\pi r^3} \]Where:

• \(d\textbf{B}\) is the infinitesimal magnetic field contributed by the current segment.
• \(I\) is the current flowing through the wire.
• \(d\textbf{l}\) is the vector length of the current element.
• \(\textbf{r}\) is the position vector from the current element to the point where the field is being calculated.
• \(r\) is the magnitude of \(\textbf{r}\).

The field calculated using this law is crucial for understanding how to solve problems regarding circular loops, as it lays the foundation for understanding magnetic fields around current-carrying conductors.

current carrying loop

In the context of electromagnetism, a current carrying loop is simply a coil or loop of wire through which current flows. This is a fundamental structure because it generates a magnetic field. The magnetic field is usually described by its strength and direction at various points in the surrounding space.

For a circular current carrying loop, the magnetic field at the center of the loop can be derived from Ampere's Law or using the Biot-Savart Law. It is given by:\[ B = \frac{\mu_0 I}{2 R} \]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 loop.
• \(R\) is the radius of the loop.

This formula reveals that the magnetic field is directly proportional to the current and inversely proportional to the loop's radius. This means that as the current increases or the radius decreases, the magnetic field at the center becomes stronger.

permeability of free space

The permeability of free space, also known as the magnetic constant, is a fundamental physical constant that is essential in the study of electromagnetism. It determines the extent to which a magnetic field can penetrate space. It is denoted by the symbol \(\mu_0\) and has a value of \(4 \pi \times 10^{-7} \, \text{T} \, \text{m/A}\).

In practical terms, this constant is used in formulas that describe the magnetic field influence produced by currents, such as in the Biot-Savart Law, which calculates the magnetic field for points away from simple current configurations.

Understanding \(\mu_0\) is key when dealing with magnetic fields in a vacuum or air, as these conditions do not significantly alter the magnetic influence created by a current.

Essentially:

• \(\mu_0\) serves as a baseline to express magnetic permeability and plays a crucial role in formulations of both the Biot-Savart Law and Ampere's Law.
• It helps quantify how much magnetic field (flux) can develop within free space due to a given electrical current.
• It bridges the relationship between magnetic field strength and current intensity.

By understanding the permeability of free space, we gain insight into the nature and strength of magnetic fields in various environments, making it a cornerstone concept in physics.