$E_y = 20 \sin(3 \times 10^6 x - 4.5 \times 10^{14} t) \text{ V/m}$
(where x, t and other values have S.I. units). The dielectric constant of the medium is _________.
(speed of light in free space is $3 \times 10^8 \text{ m/s}$)
To find the dielectric constant \( \varepsilon_r \) of the medium, we first need to calculate the phase velocity \( v \) of the wave in the medium using the given wave equation:
\( E_y = 20\sin(3 \times 10^6 x - 4.5 \times 10^{14} t) \)
Here, the wave number \( k = 3 \times 10^6 \, \text{m}^{-1} \) and the angular frequency \( \omega = 4.5 \times 10^{14} \, \text{rad/s} \). The phase velocity \( v \) is given by the relation:
\( v = \frac{\omega}{k} = \frac{4.5 \times 10^{14}}{3 \times 10^6} \)
Calculating the above expression gives:
\( v = 1.5 \times 10^8 \, \text{m/s} \)
The phase velocity in a medium is also related to its dielectric constant and the speed of light in vacuum \( c \) by:
\( v = \frac{c}{\sqrt{\varepsilon_r}} \)
Substituting \( c = 3 \times 10^8 \, \text{m/s} \) and the previously found \( v \) gives:
\( 1.5 \times 10^8 = \frac{3 \times 10^8}{\sqrt{\varepsilon_r}} \)
Solving for \( \varepsilon_r \), we square both sides:
\( (1.5)^2 = \frac{(3)^2}{\varepsilon_r} \)
\( 2.25 = \frac{9}{\varepsilon_r} \)
\( \varepsilon_r = \frac{9}{2.25} = 4 \)
The calculated dielectric constant, \( \varepsilon_r = 4 \), falls within the expected range of [4, 4], confirming the solution is correct.
Similar Questions
- A cylindrical conductor of length 2 m and area of cross-section $0.2 \text{ mm}^2$ carries an electric current of 1.6 A when its ends are connected to a 2 V battery. Mobility of electrons in the conductor is $\alpha \times 10^{-3} \text{ m}^2\text{/V.s}$. The value of $\alpha$ is :
(electron concentration $= 5 \times 10^{28} \text{/m}^3$ and electron charge = $1.6 \times 10^{-19} \text{ C}$) - A short bar magnet placed with its axis at $30^\circ$ with an external field of 800 Gauss, experiences a torque of 0.016 N.m. The work done in moving it from most stable to most unstable position is $\alpha \times 10^{-3} \text{ J}$. The value of $\alpha$ is ______.
Three parallel plate capacitors each with area $A$ and separation $d$ are filled with two dielectric ($k_1$ and $k_2$) in the following fashion. Which of the following is true? ($k_1 > k_2$)
- A moving coil galvanometer of resistance $100 \text{ }\Omega$ shows a full scale deflection for a current of $1 \text{ mA}$. The value of resistance required to convert this galvanometer into an ammeter, showing full scale deflection for a current of $5 \text{ mA}$, is _______ $\Omega$
The reading of the ammeter ($A$) in steady state in the following circuit (assuming negligible internal resistance of the ammeter) is _______ A.
- A capacitor $C$ is first charged fully with potential difference of $V_0$ and disconnected from the battery. The charged capacitor is connected across an inductor having inductance $L$. In $t\text{ s}$ $25\%$ of the initial energy in the capacitor is transferred to the inductor. The value of $t$ is __________ $\text{s}$.
The charge stored by the capacitor $C$ in the given circuit in the steady state is __________ $\mu\text{C}$.
- An electromagnetic wave of frequency $100\text{ MHz}$ propagates through a medium of conductivity, $\sigma = 10\text{ mho/m}$. The ratio of maximum conduction current density to maximum displacement current density is __________.
$\left[ \text{Take } \frac{1}{4\pi\varepsilon_0} = 9 \times 10^9\text{ Nm}^2\text{/C}^2 \right]$ - An inductor stores $16\text{ J}$ of magnetic field energy and dissipates $32\text{ W}$ of thermal energy due to its resistance when an a.c. current of $2\text{ A}$ (rms) and frequency $50\text{ Hz}$ flows through it. The ratio of inductive reactance to its resistance is _________. ($\pi = 3.14$)
- A simple pendulum made of mass 10 g and a metallic wire of length 10 cm is suspended vertically in a uniform magnetic field of 2 T. The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of $60^\circ$ with vertical, then maximum induced EMF between the point of suspension and point of oscillation is ___________ mV. (Take $g = 10\text{ m/s}^2$)
Important Questions from Electricity and Magnetism
- A cylindrical conductor of length 2 m and area of cross-section $0.2 \text{ mm}^2$ carries an electric current of 1.6 A when its ends are connected to a 2 V battery. Mobility of electrons in the conductor is $\alpha \times 10^{-3} \text{ m}^2\text{/V.s}$. The value of $\alpha$ is :
(electron concentration $= 5 \times 10^{28} \text{/m}^3$ and electron charge = $1.6 \times 10^{-19} \text{ C}$) - A short bar magnet placed with its axis at $30^\circ$ with an external field of 800 Gauss, experiences a torque of 0.016 N.m. The work done in moving it from most stable to most unstable position is $\alpha \times 10^{-3} \text{ J}$. The value of $\alpha$ is ______.
Three parallel plate capacitors each with area $A$ and separation $d$ are filled with two dielectric ($k_1$ and $k_2$) in the following fashion. Which of the following is true? ($k_1 > k_2$)
- A moving coil galvanometer of resistance $100 \text{ }\Omega$ shows a full scale deflection for a current of $1 \text{ mA}$. The value of resistance required to convert this galvanometer into an ammeter, showing full scale deflection for a current of $5 \text{ mA}$, is _______ $\Omega$
The reading of the ammeter ($A$) in steady state in the following circuit (assuming negligible internal resistance of the ammeter) is _______ A.
