An antenna is created by wrapping a square frame with side...

An antenna is created by wrapping a square frame with side length $a$ by a wire $N$ times and then connecting the leads to a resistor $R$ and a capacitor $C,$ as shown in Fig. P32.42. The loop itself has self-inductance $L .$ A plane electromagnetic wave with electric field $\overrightarrow{\boldsymbol{E}}=E_{\max } \cos (k z-\omega t) \hat{\jmath}$ propagates in the $+z$ -direction. The origin is at the center of the frame. (a) What is the magnetic flux through the coil in the direction of the $+x$ -axis?

An antenna is created by wrapping a square frame with side length $a$ by a wire $N$ times and then connecting the leads to a resistor $R$ and a capacitor $C,$ as shown in Fig. P32.42. The loop itself has self-inductance $L .$ A plane electromagnetic wave with electric field $\overrightarrow{\boldsymbol{E}}=E_{\max } \cos (k z-\omega t) \hat{\jmath}$ propagates in
the $+z$ -direction. The origin is at the center of the frame. (a) What is the magnetic flux through the coil in the direction of the $+x$ -axis?

Key Concepts

Electromagnetic Wave Polarization

Plane electromagnetic waves are transverse waves in which the electric and magnetic fields oscillate perpendicular to each other and to the direction of propagation. In the problem, the electric field is specified along the j?-direction and the wave propagates along the +z-direction. This information is used to determine the corresponding orientation of the magnetic field, which is necessary for calculating its component along the +x-axis.

Derivation of Magnetic Field Components from a Plane Wave

Given any plane electromagnetic wave, Maxwell’s equations relate the electric and magnetic fields. For example, in free space the magnetic field can be derived from the electric field using the relationship B = (1/c) k? × E, where k? is the unit vector in the direction of wave propagation. This derivation allows one to obtain the component of the magnetic field in the direction of the loop's area vector, which is essential when evaluating the magnetic flux through the coil.

Magnetic Flux

Magnetic flux is defined as the surface integral of the magnetic field over an area, mathematically expressed as ? = ? B · dA. This concept is fundamental in electromagnetism because it quantifies how much of a magnetic field passes through a given surface. The dot product emphasizes that only the component of the magnetic field normal to the surface contributes to the flux, making the orientation of the area vector critical.

Area Vector and Orientation

The area vector of a loop or surface is a vector whose magnitude is equal to the area and whose direction is perpendicular to the surface. In the context of the problem, the loop’s area vector is taken to be in the +x-direction. Understanding this concept is essential when computing magnetic flux, as it dictates which component of the magnetic field (the one parallel to the area vector) will contribute to the flux.

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