Using Faraday's law, show that if a magnetic dipole...

Using Faraday's law, show that if a magnetic dipole antenna's axis makes an angle $\theta$ with the magnetic field of an EM wave, the induced emf in the antenna is reduced from its maximum possible value by a factor of cos $\theta .$ [Hint: Assume that, at any instant, the magnetic field everywhere inside the loop is uniform. $]$

Key Concepts

Magnetic Dipole Antenna Orientation

The performance of a magnetic dipole antenna is critically dependent on its orientation with respect to the incident electromagnetic wave. When the antenna’s axis is aligned with the magnetic field, it experiences the maximum possible magnetic flux and therefore induces the maximum emf. Any deviation from this alignment results in a proportional decrease in the effective flux through the antenna, as quantified by the cosine of the misalignment angle.

Geometric Projection (Cosine Factor)

The cosine factor arises from the projection of the magnetic field onto the normal of the loop’s plane. When the magnetic dipole antenna is rotated by an angle ? relative to the magnetic field, only the component of the field aligned with the antenna’s axis (i.e., the normal component) contributes to the magnetic flux. Hence, the induced emf is reduced by a factor of cos ?, reflecting the geometric alignment between the field and the antenna.

Faraday's Law

Faraday's Law states that a time-varying magnetic flux through a closed loop induces an electromotive force (emf) in the loop. This principle governs how the changing magnetic field of an electromagnetic wave can induce a voltage in a magnetic dipole antenna, linking the variation of the magnetic field to the resulting electrical response.

Magnetic Flux

Magnetic flux is the measure of the magnetic field passing through a given area, quantified as the product of the magnetic field strength, the area of the surface, and the cosine of the angle between the field direction and the normal to the surface. It is a key parameter in determining the induced emf since the rate of change of flux is what drives the induction process.

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