then the refractive index of the medium is ________.
(All given measurement are in SI units)
Key Concept: Refractive Index Calculation
The refractive index ($n$) of a medium is defined as the ratio of the speed of light in vacuum ($c$) to the speed of light in the medium ($v$), i.e., $n = c/v$. The speed of light ($v$) in a medium through which an electromagnetic wave propagates is related to the wave's angular frequency ($\omega$) and wave number ($k$) by the formula $v = \omega/k$.
Extracting Wave Parameters
The electric field of the electromagnetic wave is given by $\vec{E}(x,t) = 25 \sin(2.0 \times 10^{15} t - 10^7 x) \hat{n}$. By comparing this to the general form $\vec{E}(x,t) = E_0 \sin(\omega t - kx)$, we can identify the parameters:
- Angular frequency, $\omega = 2.0 \times 10^{15}$ rad/s.
- Wave number, $k = 10^7$ m$^{-1}$.
- The speed of light in vacuum is a constant, $c \approx 3.0 \times 10^8$ m/s.
Calculating Refractive Index
First, calculate the speed of light ($v$) in the medium using the identified $\omega$ and $k$:
$ v = \frac{\omega}{k} $
Substituting the values:
$ v = \frac{2.0 \times 10^{15} \text{ rad/s}}{10^7 \text{ m}^{-1}} = 2.0 \times 10^{8} \text{ m/s} $
Next, calculate the refractive index ($n$) using the calculated speed ($v$) and the speed of light in vacuum ($c$):
$ n = \frac{c}{v} $
Substituting the values of $c$ and $v$:
$ n = \frac{3.0 \times 10^8 \text{ m/s}}{2.0 \times 10^{8} \text{ m/s}} = 1.5 $
Thus, the refractive index of the medium is 1.5.
Similar Questions
- The electric current in the circuit is given as $i = i_o(t/T)$. The r.m.s current for the period $t = 0$ to $t = T$ is ________.
- In the potentiometer, when the cell in the secondary circuit is shunted with $4\text{ }\Omega$ resistance, the balance is obtained at the length $120\text{ cm}$ of wire. Now when the same cell is shunted with $12\text{ }\Omega$ resistance, the balance is shifted to a length of $180\text{ cm}$. The internal resistance of cell is ________$\Omega$
Three long straight wires carrying current are arranged mutually parallel as shown in the figure. The force experienced by $15 \text{ cm}$ length of wire $Q$ is________.
$(\mu_o = 4\pi \times 10^{-7} \text{ T.m/A})$- For the two cells having same EMF $E$ and internal resistance $r$, the current passing through the external resistor $6\text{ }\Omega$ is same when both the cells are connected either in parallel or in series. The value of internal resistance $r$ is ________$\Omega$.
- The magnetic field at the centre of a current carrying circular loop of radius $R$ is $16 \text{ \mu T}$. The magnetic field at a distance $x = \sqrt{3}R$ on its axis from the centre is ________$\text{\mu T}$.
- Two point charges of $1 \text{ nC}$ and $2 \text{ nC}$ are placed at the two corners of equilateral triangle of side $3 \text{ cm}$. The work done in bringing a charge of $3 \text{ nC}$ from infinity to the third corner of the triangle is ________$\text{\mu J}$.
$\frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ N.m}^2\text{/C}^2$ The equivalent resistance between the points $A$ and $B$ in the following circuit is $\frac{x}{5} \text{ }\Omega$. The value of $x$ is ________.
- Three identical coils $C_1, C_2$ and $C_3$ are closely placed such that they share a common axis. $C_2$ is exactly midway. $C_1$ carries current $I$ in anti-clockwise direction while $C_3$ carries current $I$ in clockwise direction. An induced current flows through $C_2$ will be in clockwise direction when
A meter bridge with two resistances $R_1$ and $R_2$ as shown in figure was balanced (null point) at 40 cm from the point $P$. The null point changed to 50 cm from the point $P$, when $16 \ \Omega$ resistance is connected in parallel to $R_2$. The values of resistances $R_1$ and $R_2$ are _________.
XPQY is a vertical smooth long loop having a total resistance $R$ where PX is parallel to QY and separation between them is $l$. A constant magnetic field $B$ perpendicular to the plane of the loop exists in the entire space. A rod CD of length $L \ (L > l)$ and mass $m$ is made to slide down from rest under the gravity as shown in figure. The terminal speed acquired by the rod is _________ m/s. (g = acceleration due to gravity)
Important Questions from Electricity and Magnetism
- The electric current in the circuit is given as $i = i_o(t/T)$. The r.m.s current for the period $t = 0$ to $t = T$ is ________.
- In the potentiometer, when the cell in the secondary circuit is shunted with $4\text{ }\Omega$ resistance, the balance is obtained at the length $120\text{ cm}$ of wire. Now when the same cell is shunted with $12\text{ }\Omega$ resistance, the balance is shifted to a length of $180\text{ cm}$. The internal resistance of cell is ________$\Omega$
Three long straight wires carrying current are arranged mutually parallel as shown in the figure. The force experienced by $15 \text{ cm}$ length of wire $Q$ is________.
$(\mu_o = 4\pi \times 10^{-7} \text{ T.m/A})$- For the two cells having same EMF $E$ and internal resistance $r$, the current passing through the external resistor $6\text{ }\Omega$ is same when both the cells are connected either in parallel or in series. The value of internal resistance $r$ is ________$\Omega$.
- The magnetic field at the centre of a current carrying circular loop of radius $R$ is $16 \text{ \mu T}$. The magnetic field at a distance $x = \sqrt{3}R$ on its axis from the centre is ________$\text{\mu T}$.
