Which of the following relations is correct?

Question

Examoogle · Accepted Answer

Option C: $Emf = \int \vec{E} \cdot d\vec{l}$

Explanation & Concepts:
Electromotive Force (EMF) is defined as the work done per unit charge in moving a charge along a closed path, which is mathematically represented by the closed line integral of the electric field intensity vector $\vec{E}$ along the contour $l$. By Faraday's law of induction, the induced EMF in a loop is equal to the negative rate of change of magnetic flux linkage.

Working Principle: According to Maxwell-Faraday equation in integral form, $\oint_C \vec{E} \cdot d\vec{l} = -\frac{d}{dt} \iint_S \vec{B} \cdot d\vec{S}$. This indicates that a time-varying magnetic field induces an electric field, creating a potential difference (EMF) around a closed path. The quantity $\int \vec{E} \cdot d\vec{l}$ represents the potential energy generated per unit charge.

Key Formulas: $EMF = \oint_C \vec{E} \cdot d\vec{l}$ — Definition of EMF as circulation of Electric Field; $MMF = \oint_C \vec{H} \cdot d\vec{l}$ — Definition of Magnetomotive Force as circulation of Magnetic Field intensity

Key Points: EMF is the scalar potential difference per unit charge.; The line integral of the Electric Field intensity $\vec{E}$ represents the work done by the field.; Faraday's Law relates the induced EMF to the change in magnetic flux.; MMF corresponds to the line integral of magnetic field intensity $\vec{H}$ (Ampere's Circuital Law).

Advantages: Provides a fundamental understanding of voltage generation.; Essential for analyzing transformer and induction motor operations.

Disadvantages: Requires knowledge of vector calculus.; Only applies to closed loops in time-varying fields.

Applications: Electrical machine design (generators and motors).; Transformer winding voltage analysis.; Electromagnetic compatibility studies.

Additional Info: Option A is incorrect because the line integral of the Magnetic Flux Density $\vec{B}$ is not defined as MMF; the correct relation for MMF is the line integral of the Magnetic Field Intensity $\vec{H}$.. Option D is incorrect as $\vec{D}$ is the Electric Flux Density, and its line integral does not define EMF.

Answer: C is correct — EMF is defined as the circulation of the electric field intensity vector around a closed path.

Exam Tip: Always remember that EMF correlates with the Electric Field ($\vec{E}$), while MMF correlates with the Magnetic Field Intensity ($\vec{H}$).

Examoogle · Answer

Option A: $Mmf = \int \vec{B} \cdot d\vec{l}$

Examoogle · Answer

Option B: $Mmf = \int \vec{H} \cdot d\vec{l}$

Examoogle · Answer

Option D: $Emf = \int \vec{D} \cdot d\vec{l}$

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