$E_y = 20 \sin(3 \times 10^6 x - 4.5 \times 10^{14} t) \text{ V/m}$
(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
Figure shows the circuit that contains three resistances ($9 \, \Omega$ each) and two inductors (4 mH each). The reading of ammeter at the moment switch K is turned ON, is _________ A.
- A capacitor $P$ with capacitance $10 \times 10^{-6} \text{ F}$ is fully charged with a potential difference of 6.0 V and disconnected from the battery. The charged capacitor $P$ is connected across another capacitor $Q$ with capacitance $20 \times 10^{-6} \text{ F}$. The charge on capacitor $Q$ when equilibrium is established will be $\alpha \times 10^{-5} \text{ C}$ (assume capacitor $Q$ does not have any charge initially), the value of $\alpha$ is _________.
- A conducting circular loop is rotated about its diameter at a constant angular speed of 100 rad/s in a magnetic field of 0.5 T perpendicular to the axis of rotation. When the loop is rotated by $30^\circ$ from the horizontal position, the induced EMF is 15.4 mV. The radius of the loop is _________ mm.
$\left(\text{Take } \pi = \frac{22}{7}\right)$ - 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 = $1.6 \times 10^{-19} \text{ C}$) - There are three co-centric conducting spherical shells $A$, $B$ and $C$ of radii $a$, $b$ and $c$ respectively ($c > b > a$) and they are charged with charge $q_1$, $q_2$ and $q_3$ respectively. The potentials of the spheres $A$, $B$ and $C$ respectively, are :
For the series $LCR$ circuit connected with 220 V, 50 Hz a.c source as shown in the figure, the power factor is $\frac{\alpha}{10}$. The value of $\alpha$ is ______.
- Two resistors of $100\, \Omega$ each are connected in series with a 9 V battery. A voltmeter of $400\, \Omega$ resistance is connected to measure the voltage drop across one of the resistors. The voltmeter reading is ______ V.
- The electrostatic potential in a charged spherical region of radius $r$ varies as $V = ar^3 + b$, where $a$ and $b$ are constants. The total charge in the sphere of unit radius is $\alpha \times \pi a \epsilon_0$. The value of $\alpha$ is ______.
(permittivity of vacuum is $\epsilon_0$) Two resistors $2\, \Omega$ and $3\, \Omega$ are connected in the gaps of bridge as shown in figure. The null point is obtained with the contact of jockey at some point on wire $XY$. When an unknown resistor is connected in parallel with $3\, \Omega$ resistor, the null point is shifted by 22.5 cm toward $Y$. The resistance of unknown resistor is ______ $\Omega$.
Match the LIST-I with LIST-II
List-I: List-II: A. Radio-wave I. is produced by Magnetron valve B. Micro-wave II. due to change in the vibrational modes of atoms C. Infrared-wave III. due to inner shell electrons moving from higher energy level to lower energy level D. X-ray IV. due to rapid acceleration of electrons Choose the correct answer from the options given below:
Important Questions from Electricity and Magnetism
Figure shows the circuit that contains three resistances ($9 \, \Omega$ each) and two inductors (4 mH each). The reading of ammeter at the moment switch K is turned ON, is _________ A.
- A capacitor $P$ with capacitance $10 \times 10^{-6} \text{ F}$ is fully charged with a potential difference of 6.0 V and disconnected from the battery. The charged capacitor $P$ is connected across another capacitor $Q$ with capacitance $20 \times 10^{-6} \text{ F}$. The charge on capacitor $Q$ when equilibrium is established will be $\alpha \times 10^{-5} \text{ C}$ (assume capacitor $Q$ does not have any charge initially), the value of $\alpha$ is _________.
- A conducting circular loop is rotated about its diameter at a constant angular speed of 100 rad/s in a magnetic field of 0.5 T perpendicular to the axis of rotation. When the loop is rotated by $30^\circ$ from the horizontal position, the induced EMF is 15.4 mV. The radius of the loop is _________ mm.
$\left(\text{Take } \pi = \frac{22}{7}\right)$ - 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 = $1.6 \times 10^{-19} \text{ C}$) - There are three co-centric conducting spherical shells $A$, $B$ and $C$ of radii $a$, $b$ and $c$ respectively ($c > b > a$) and they are charged with charge $q_1$, $q_2$ and $q_3$ respectively. The potentials of the spheres $A$, $B$ and $C$ respectively, are :
