Why does microwave oven heats up a food item containing water molecules most efficiently?
Microwave oven heats up the food items containing water molecules most efficiently because the frequency of microwaves matches the resonant frequency of water molecules.
The charge on a parallel plate capacitor varies as $q=q_0 \cos 2 \pi v t$. The plates are very large and close together (area $=A$, separation $=d$ ). Neglecting the edge effects, find the displacement current through the capacitor.
The displacement current through the capacitor is,
$$I_d=I_c=\frac{d q}{d t}\quad\text{... (i)}$$
Here, $$\quad q=q_0 \cos 2 \pi v t \text { (given) }$$
Putting this value in Eq (i), we get
$$\begin{aligned} & I_d=I_c=-q_0 \sin 2 \pi v t \times 2 \pi v \\ & I_d=I_c=-2 \pi v q_0 \sin 2 \pi v t \end{aligned}$$
'A variable frequency AC source is connected to a capacitor. How will the displacement current change with decrease in frequency?
$$\begin{aligned} & \text { Capacitive reaction } X_c=\frac{1}{2 \pi f C} \\ & \therefore \quad X_c \propto \frac{1}{f} \end{aligned}$$
As frequency decreases, $X_{\mathrm{c}}$ increases and the conduction current is inversely proportional to $X_c\left(\because I \propto \frac{1}{X_c}\right)$.
So, displacement current decreases as the conduction current is equal to the displacement current.
The magnetic field of a beam emerging from a filter facing a floodlight is given by
$$B_0=12 \times 10^{-8} \sin \left(1.20 \times 10^7 z-3.60 \times 10^{15} t\right) \mathrm{T}$$
What is the average intensity of the beam?
Magnetic field $\mathbf{B}=B_0$ sin $\omega t$
Given, equation $B=12 \times 10^{-8} \sin \left(1.20 \times 10^7 z-3.60 \times 10^{15} t\right) \mathrm{T}$.
On comparing this equation with standard equation, we get
$$B_0=12 \times 10^{-8}$$
The average intensity of the beam $I_{\mathrm{av}}=\frac{1}{2} \frac{B_0^2}{\alpha_0} \cdot c=\frac{1}{2} \times \frac{\left(12 \times 10^{-8}\right)^2 \times 3 \times 10^8}{4 \pi \times 10^{-7}}$
$$=1.71 \mathrm{~W} / \mathrm{m}^2$$
Poynting vectors $\mathbf{S}$ is defined as a vector whose magnitude is equal to the wave intensity and whose direction is along the direction of wave propogation. Mathematically, it is given by $\mathbf{S}=\frac{1}{\propto_0} \mathbf{E} \times \mathbf{B}$. Show the nature of $\mathbf{S}$ versus $t$ graph.
Consider and electromagnetic waves, let $\mathbf{E}$ be varying along $y$-axis, $\mathbf{B}$ is along $z$-axis and propagation of wave be along $x$-axis. Then $\mathbf{E} \times \mathbf{B}$ will tell the direction of propagation of energy flow in electromegnetic wave, along $x$-axis.
Let
$$\begin{aligned} \mathbf{E} & =E_0 \sin (\omega t-k x) \hat{\mathbf{j}} \\ \mathbf{B} & =B_0 \sin (\omega t-k x) \hat{\mathbf{k}} \\ \mathbf{S} & =\frac{1}{\alpha_0}(\mathbf{E} \times \mathbf{B})=\frac{1}{\alpha_0} E_0 B_0 \sin ^2(\omega t-k x)[\hat{\mathbf{j}} \times \hat{\mathbf{k}}] \\ & =\frac{E_0 B_0}{\alpha_0} \sin ^2(\omega t-k x) \hat{\mathbf{i}} \end{aligned}$$
The variation of $|\mathbf{S}|$ with time $t$ will be as given in the figure below
