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. $\mathbf{P 3 2 . 4 2 .}$ The loop itself has self-inductance $L$. $\mathrm{A}$ 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? (Hint: The identity $\sin (A+B)+\sin (A-B)=2 \sin A \cos B$ may prove helpful.) (b) What is the emf generated in the coil? (c) The electromagnetic wave has frequency $4.00 \mathrm{Mz}$ and intensity $100 \mathrm{~W} / \mathrm{m}^{2} .$ The coil has $N=50$ windings and side length $a=10.0 \mathrm{~cm} .$ It follows that its selfinductance is $L=78.0 \mu H$. If the resistance in the circuit is $R=100 \Omega$, what value of the capacitance $C$ results in the resonance frequency of the $L-R-C$ circuit being equal to the frequency of the wave? (d) What rms value of the current $I_{\text {rms }}$ flows in that case?

Key Concepts

Plane Electromagnetic Waves

Plane electromagnetic waves are solutions to Maxwell's equations characterized by sinusoidally varying electric and magnetic fields that are perpendicular to each other and propagate in a uniform direction. Understanding their behavior is vital in analyzing interactions with antenna systems and circuits.

RMS Values in AC Circuits

The RMS (root mean square) value is a measure used to express the effective magnitude of an alternating current or voltage. It is particularly useful because it relates directly to the power delivered in a resistive load, allowing AC values to be compared with equivalent direct current values.

Resonant RLC Circuits

Resonant RLC circuits involve resistors, inductors, and capacitors arranged such that at a particular frequency, known as the resonance frequency, the inductive and capacitive reactances cancel each other. This results in a circuit with minimal impedance and maximal current response, essential in applications like tuning and filtering.

Magnetic Flux

Magnetic flux represents the total magnetic field passing normally through a surface. It is calculated as the dot product of the magnetic field and the area vector, and variations in flux over time are central to the process of electromagnetic induction.

Faraday's Law of Induction

Faraday's Law of Induction states that any change in the magnetic environment of a coil will induce an electromotive force (emf) in the coil. This is the foundational principle behind the operation of transformers, motors, and many types of antennas where time-varying magnetic fields produce currents.

Self-Inductance

Self-inductance is the property of a circuit (particularly a coil) that quantifies its tendency to oppose changes in current due to the magnetic flux created by the current itself. It is a key parameter in analyzing how coils and inductors behave in time-varying electromagnetic environments.

Transcript

00:01 In this question, we have an antenna.

00:03 Okay, the antenna looks like this.

00:06 So, okay, it's a square loop, okay, with entrance, okay, entrance, and then this is 10 cm, and, okay, this is also 10 cm.

00:33 Okay, and then this is connected with a resistor, and then with a capacitor.

00:40 And then there's a current that is flowing in the circuit.

00:50 It has an l, the self -inductance.

00:55 There's something that we need later.

00:58 Okay, so in this part is about how an antenna collect information or energy from electromagnetic wave and then from the energy that it collects from the energy that it collects from the energy.

01:13 The antenna it drives the circuit and then we want to find out the magnetic flux induce the mf the capacitance for resonant frequency and the rms value for the current and slowing in the circuit okay so to do this question okay in part a we want to find a magnetic flux okay we are given the the em wave.

01:47 Okay, it looks like this.

01:50 The electric view is emax, cosine, kz minus omega -t, j hats, and then we are given that the coordinate system.

02:08 It looks like this.

02:10 This is x, this is y this is z okay based on the given expression the wave is propagating along the plus z head direction and with e in the j with e in j hat okay then we need our b then b is in negative i hat so that e cross b gives z heads okay so which is the propagation of propagation of the em wave okay so our magnetic field vector is b equals to b max cosine k z minus omega t in the negative i head direction okay and then we know that c b max is equal to e max okay there's something with that we need later okay so in part a you want to find a magnetic flux so the definition of magnetic flux is b.

03:41 D a okay so the d a i'm going to make my d a to be in the negative i head direction.

03:59 Okay, so the a is going to be from the diagram.

04:07 Okay, so the b field is pointing in this direction.

04:14 Okay, and then i'm going to set my da in this direction.

04:17 And then my da will be expressed in terms of d .a.

04:23 D z okay it will be a times d z okay so a times d z um negative i had okay so by doing so then my uh magnetic flux is going to be positive okay so uh b max um cosy k z minus omega t times a d z okay okay integrate from minus a over 2 to a over 2 okay the origin of the coordinate system is at the center of the coin okay so take out the b max and a since they are the constants you integrate cosine you get sign and then because you are integrating over z respect to z so you get one over k x x minus omega t okay a over two integrating from negative a over two to a over two okay so b max a over k times si k a over two minus omega t minus si si x minus k a over two minus k a over two minus k a over two minus omega t okay so you simplify further okay so the second expression note that sine negative x is equal to negative sine x okay so we can pull out the negative sign and you get a plus and then after that you use a given identity okay so the given identity is um sign a plus b plus sine a minus b okay is equal to two sine a cosine cosine b okay so we can actually simplify this whole thing as two sine k a over two uh cosine omega t okay so this will be the 2b max a over k x x x k a over 2 cosy omega t so this is the magnetic flux okay so this is our answer for part a next in part b you want to calculate the induce mf so to calculate the induce mf using for this law okay we say that d induced is equal to negative defy b t yes there's one thing i forgot to add there's a n okay in each step okay so what i calculated is the many flux for one coin so if you need and coins that means be two times n k okay so the einduce is negative dd t of the magnetic flux which is 2n b max a over k, si k a over 2, cosine, cosine omega t.

08:42 Okay and you differentiate there's only the one thing that is as a function of time.

08:49 So differentiate cosine you get negative sine so and then you also get that omega out okay and you see that omega of k is c okay to n c b max uh times a okay times x k x k a over 2 sine k omega t okay so this is the the emf generated okay, next in part c you want to find the capacitance for resonance to take place.

09:43 Okay, so we are given the frequency is 4 megahertz...

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