The average power of the electromagnetic wave is $\left(\frac{1}{\alpha}\right)\text{ W/m}^2$. The value of $\alpha$ is ___________.
$\left[ \text{Take } \sqrt{\frac{\mu_0}{\epsilon_0}} = 377\text{ in SI units} \right]$
Objective: Calculate Alpha from EM Wave Power
The question asks for the value of $\alpha$ given the electric field ($E$) of an electromagnetic wave and its average power expressed as $\left(\frac{1}{\alpha}\right) \text{ W/m}^2$. The wave propagates through free space.
Electromagnetic Wave Properties
The equation for the electric field is given as: $E = \sqrt{377} \sin(6.27 \times 10^3 t - 2.09 \times 10^{-5} x)\text{ N/C}$
- From the standard wave equation $E = E_0 \sin(\omega t - kx)$, the amplitude of the electric field is $E_0 = \sqrt{377} \text{ N/C}$.
- The wave is propagating in free space. The intrinsic impedance of free space is $Z_0 = \sqrt{\frac{\mu_0}{\epsilon_0}} = 377 \text{ } \Omega$.
Calculating Average Power (Intensity)
The average power per unit area (Intensity, $I$) of an electromagnetic wave in free space is related to the electric field amplitude ($E_0$) and the impedance ($Z_0$) by the formula:
$I = \frac{1}{2} \frac{E_0^2}{Z_0}$Substitute the values:
$I = \frac{1}{2} \frac{(\sqrt{377})^2}{377}$ $I = \frac{1}{2} \frac{377}{377}$ $I = \frac{1}{2} = 0.5 \text{ W/m}^2$Determining the Value of Alpha
The problem states that the average power is $\left(\frac{1}{\alpha}\right) \text{ W/m}^2$. We have calculated the intensity (average power) $I$ to be $0.5 \text{ W/m}^2$. Therefore:
$\frac{1}{\alpha} = I$ $\frac{1}{\alpha} = 0.5$Solving for $\alpha$:
$\alpha = \frac{1}{0.5}$ $\alpha = 2$The value of $\alpha$ is 2.
Similar Questions
- 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 ______.
- 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 $g = 10\text{ m/s}^2$)
Using a variable frequency a.c. voltage source the maximum current measured in the given LCR circuit is 50 mA for $V = 5\sin(100t)$ The values of L and R are shown in the figure. The capacitance of the capacitor (C) used is ___________ $\mu\text{F}$.
The space between the plates of a parallel plate capacitor of capacitance C (without any dielectric) is now filled with three dielectric slabs of dielectric constants $K_1 = 2$, $K_2 = 3$ and $K_3 = 5$ (as shown in figure). If new capacitance is $\frac{n}{3}C$ then the value of $n$ is ___________.
Combination of four parallel plate air capacitors is shown in figure. Separation between the plate P1 & P2 is $\frac{d}{3}$ and separation between the plates P2 and P3 is d.
The equivalent capacitance of the configuration is _____________$\mu$F.
(Given $\frac{\epsilon_0 A}{d} = 4\mu$F . Where A=Area of plates)
- A plane polarized monochromatic EM wave is traveling in vacuum along z direction such that at $t = t_1$ it is found that the electric field is zero at a spatial point $z_1$. The next zero that occurs in its neighbourhood is at $z_2$. The frequency of the electromagnetic wave is :
A copper rod of mass m slides under gravity on two smooth parallel rails, with separation $l$ and set at an angle of $\theta$ with the horizontal. At the bottom, rails are joined by a resistance R. There is a uniform magnetic field B normal to the plane of the rails, as shown in the figure. The terminal speed of the copper rod is :
- A copper rod of cross-sectional area A carries a uniform current I through it. At temperature T, if the volume charge density of the rod is $\rho$, how long will the charges take to travel a distance d?
Important Questions from Electricity and Magnetism
- 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 ______.
- 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 $g = 10\text{ m/s}^2$)
Using a variable frequency a.c. voltage source the maximum current measured in the given LCR circuit is 50 mA for $V = 5\sin(100t)$ The values of L and R are shown in the figure. The capacitance of the capacitor (C) used is ___________ $\mu\text{F}$.
