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Animated Solution for Physics - Magnetic Effects of Current: A circular loop of radius , carrying current , lies in - plane with its centre at origin. The total magnetic flux through - plane is

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Visualized Solution

\text{The Setup}

  • \text{Circular loop in } x-y \text{ plane}
  • \text{Radius } = R
  • \text{Current } = I

\text{Magnetic Field Lines}

  • \text{Property of Magnetic Field:}
  • \nabla \cdot \mathbf{B} = 0
  • \text{Field lines form closed loops.}

\text{Flux Inside the Loop}

  • \text{Inside the loop:}
  • \mathbf{B} \text{ is in } +z \text{ direction}
  • \Phi_{\text{inside}} > 0

\text{Flux Outside the Loop}

  • \text{Outside the loop:}
  • \mathbf{B} \text{ is in } -z \text{ direction}
  • \Phi_{\text{outside}} < 0

\text{Total Magnetic Flux}

  • \text{Since every line that goes up must come down:}
  • |\Phi_{\text{inside}}| = |\Phi_{\text{outside}}|
  • \Phi_{\text{total}} = \Phi_{\text{inside}} + \Phi_{\text{outside}} = 0

\text{Conclusion}

  • \text{Final Answer: } 0
  • \text{Note: Flux through the loop itself is non-zero.}

The Sigma Insight: Magnetic Field

Solution Diagram
The concept of magnetic flux often brings to mind complex surface integrals and challenging dot products. However, some problems are elegantly solved not with heavy mathematics, but with a profound understanding of fundamental physical laws.

Analyzing the Setup

Imagine an infinite - plane. Placed perfectly within this plane is a circular loop of radius , carrying a steady current . The question asks for the total magnetic flux through this entire infinite - plane.
At first glance, you might be tempted to calculate the magnetic field at a distance from the center and integrate it from to . While mathematically possible, it is a trap! There is a much more beautiful and intuitive way to arrive at the answer.

The Master Principle

Closed Loops
To solve this, we must recall the most fundamental property of magnetic field lines: Magnetic field lines always form continuous, closed loops. Unlike electric field lines, which can originate from a positive charge and terminate on a negative charge, magnetic monopoles do not exist.
Let's apply the right-hand thumb rule to our current-carrying loop. If the current flows counter-clockwise, the magnetic field lines inside the loop will emerge outwards, pointing in the direction.

The Balancing Act

Here is where the magic happens. Because these magnetic field lines must form closed loops, every single line that emerges from the inside of the loop must eventually curve around and dive back into the - plane in the region outside the loop.
When they dive back in, they point in the direction.
Since every line that goes up must come down, the total number of lines crossing the plane in the positive direction is exactly equal to the total number of lines crossing the plane in the negative direction.

Final Calculation

When we sum the flux over the entire infinite - plane, the positive flux from the inside region perfectly cancels the negative flux from the outside region.
The total magnetic flux through the entire - plane is exactly zero. This is a classic JEE conceptual trap—always read carefully whether the question asks for the flux through the loop or through the entire plane!