A circular loop of radius $ r $ is placed in a uniform magneti... | General Science MCQ | Examoogle

General SciencePhysics: Electromagnetism

A circular loop of radius $r$ is placed in a uniform magnetic field $\vec{B}$ . If the plane of the loop is perpendicular to the field, the magnetic flux through the loop is ______.

$0$

$\pi r^2 B$

$2\pi r B$

$\pi r B$

Correct Answer

🔬 SC • Science Concept & PrincipleGeneral SciencePhysics: Electromagnetism

Option C

$2\pi r B$

Quick Summary:

The magnetic flux $\Phi_B$ through a surface is defined as the product of the magnetic field strength $B$ and the area $A$ perpendicular to the field. For a circular loop of radius $r$ oriented perpendicular to a uniform magnetic field, the area vector is parallel to the magnetic field vector, resulting in $\Phi_B = B \times A = B \times (\pi r^2)$ .

🔬SCScience SolutionConcept & Principle

💡 Explanation

💡 Everyday Analogy (Real-World Intuition)

Think of a window screen catching rain; if the screen is tilted, it catches less rain, but when it is fully perpendicular to the falling rain, it catches the maximum amount of water.

🧠 Memory Mnemonic / Shortcut Aid

Flux is 'BA-cos', where 'BA' is the area of the circular 'cos'metic mirror.

🔢 Key Formulas

$\Phi_B = \vec{B} \cdot \vec{A} = BA \cos \theta$ — General magnetic flux formula

$A = \pi r^2$ — Area of a circular loop

⚙️ Working Principle

The magnetic flux is defined by the dot product $\Phi_B = \int \vec{B} \cdot d\vec{A}$ . When the plane of the loop is perpendicular to the field, the angle $\theta$ between the area vector (normal to the loop) and the magnetic field vector is $0°$ . Since $\cos(0°) = 1$ , the flux simplifies to $\Phi_B = B \cdot A = B(\pi r^2)$ .

📌 Key Points

▸
The area vector is always normal (perpendicular) to the surface of the loop.
▸
Magnetic flux is a scalar quantity measured in Weber ( $Wb$ ).
▸
Maximum flux occurs when the field lines are perpendicular to the plane of the coil ( $\theta = 0°$ for the area vector).

✅ Advantages

▸
Useful for calculating induced EMF via Faraday's Law

❌ Disadvantages / Limitations

▸
Valid only for uniform magnetic fields and flat loops

🛠️ Applications / Uses

▸
Electric generators
▸
Induction motors

📄 Additional Information

▸
The official answer key provided in the prompt is incorrect as it specifies $2\pi r B$ , which is the formula for the circumference multiplied by $B$ , not the area flux.
▸
Option B ( $B \pi r^2$ ) is the physically correct expression for magnetic flux through a circular loop placed perpendicular to the field.

📊 Diagram / Illustration

✅

B is correct — The magnetic flux is calculated as the product of the magnetic field and the area of the loop ( $B \times \pi r^2$ ).

Core Concepts Used

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Magnetic Flux Electromagnetic Induction Vector Area

💡 EXAM TIP

In competitive exams, always check the orientation of the 'area vector' vs the 'plane of the loop' to avoid $\sin$ vs $\cos$ confusion.

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