Question

A circular loop of radius \(\mathrm{R}\) carrying current I lies in \(\mathrm{X}-\mathrm{Y}\) plane with its centre at origin. The total magnetic flux through \(\mathrm{X}-\mathrm{Y}\) plane is...... (a) Directly proportional to I (b) Directly proportional to \(\mathrm{R}\) (c) Directly proportional to \(\mathrm{R}^{2}\) (d) Zero

Short answer

The total magnetic flux through the XY plane for a circular loop of radius R carrying current I is zero.

Step-by-step solution

Expression for Magnetic Field of a Circular Loop

We know that the magnetic field due to a circular loop carrying current I at a perpendicular distance x from the center of the loop is given by the expression: \[B(x) = \frac{\mu_0 I R^2}{2 \pi (R^2 + x^2)^{3/2}}\] Where, \(B(x)\) = Magnetic field at a perpendicular distance x from the center of the loop \(\mu_0\) = Permeability of free space I = Current in the loop R = Radius of the circular loop

Calculate the Magnetic Flux

The magnetic flux (Φ) through the XY plane is the integral of the magnetic field over the XY plane area: \[\Phi = \int\int_{XY} B(x) \, dA\] Considering the symmetry of the problem, the magnetic flux through each small elemental area (dA) will be: \(d\Phi = B(x) \, dA\) Now integrating the flux, for the entire XY plane (from x = -∞ to x = +∞): \[\Phi = \int\limits_{-∞}^{+∞} B(x) \, dA = \int\limits_{-∞}^{+∞} \frac{\mu_0 I R^2}{2 \pi (R^2 + x^2)^{3/2}} \, dA\]

Determine the Magnetic Flux for the entire XY plane

We notice that the integrand contains the term \((R^2 + x^2)^{3/2}\) in the denominator, which means that as x goes from -∞ to +∞, the value of this term in the denominator will approach infinity, making the entire integrand approach zero. Therefore, the magnetic flux through the XY plane will be zero: \[\Phi = 0\] Hence, the correct answer is: (d) Zero

Key concepts

Circular Loop

A circular loop is a shape that's perfectly round, like a circle. When it comes to physics, particularly electromagnetism, a circular loop is often used to carry an electric current, denoted as I. Imagine this loop lying flat in the XY plane, centered at the origin of a coordinate system.
This setup is very common in electromagnetism problems because it allows for symmetrical analysis.

The radius of the circular loop, labeled as R, determines the size of the loop. The characteristics of the magnetic field produced by the loop are directly affected by both the current and this radius.

• Radius (R): The distance from the center of the circle to any point on its circumference.
• Current (I): The flow of electric charge through the loop.

The symmetry of a circular loop means it simplifies calculations, especially when determining magnetic fields and flux, which are central to many physics problems.

Magnetic Field

A magnetic field is an invisible field that exerts force on substances that are sensitive to magnetism.
In the context of a circular loop carrying a current I in a plane, the magnetic field is created by the movement of the electric charge.

The strength of this magnetic field depends on several factors:

• The amount of current (I) flowing through the loop.
• The radius (R) of the loop, since a larger loop will distribute the magnetic field over a larger area.

The magnetic field at a perpendicular distance x from the center of the loop can be calculated using the formula:
\[ B(x) = \frac{\mu_0 I R^2}{2 \pi (R^2 + x^2)^{3/2}} \]
Where \( \mu_0 \) is the permeability of free space. This formula shows that the magnetic field strength decreases as the distance from the loop increases due to the \((R^2 + x^2)^{3/2}\) term in the denominator.

Current

Current is the flow of electric charge across a given point or surface within a conductor, like a wire.
It's essentially the movement of electrons within the loop that generates a magnetic field.

• Measured in Amperes (A): The unit of current is the Ampere, describing how much charge is moving past a point in a second.
• Direction Matters: The direction of the current affects the direction of the generated magnetic field, which follows the right-hand rule.

When current flows through a circular loop, it continuously circulates, creating a magnetic field perpendicular to the plane of the loop.
By understanding the relationship between current and magnetic fields, you can solve many problems in electromagnetism, especially when assessing the magnetic flux in specific regions.