$\left[ \text{Take } \frac{1}{4\pi\varepsilon_0} = 9 \times 10^9\text{ Nm}^2\text{/C}^2 \right]$
Conducting Medium Current Density Ratio Calculation
The problem asks for the ratio of the maximum conduction current density ($J_{c,max}$) to the maximum displacement current density ($J_{d,max}$) for an electromagnetic wave in a conducting medium.
Given Information
- Frequency, $f = 100 \text{ MHz} = 100 \times 10^6 \text{ Hz} = 1 \times 10^8 \text{ Hz}$
- Conductivity, $\sigma = 10 \text{ mho/m}$
- Coulomb's constant, $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9 \text{ Nm}^2/\text{C}^2$
Key Concepts and Formulas
- The conduction current density is given by $J_c = \sigma E$.
- The displacement current density is $J_d = \varepsilon \frac{\partial E}{\partial t}$.
- For a time-harmonic electric field $E(t) = E_{max} \cos(\omega t)$, we have $\frac{\partial E}{\partial t} = -\omega E_{max} \sin(\omega t)$. The maximum magnitude of $\frac{\partial E}{\partial t}$ is $\omega E_{max}$.
- Therefore, the maximum conduction current density is $|J_{c,max}| = \sigma E_{max}$.
- The maximum displacement current density is $|J_{d,max}| = \omega \varepsilon E_{max}$.
- The ratio required is $\frac{|J_{c,max}|}{|J_{d,max}|} = \frac{\sigma E_{max}}{\omega \varepsilon E_{max}} = \frac{\sigma}{\omega \varepsilon}$.
Calculation Steps
- Calculate the angular frequency ($\omega$):
$\omega = 2\pi f = 2\pi (1 \times 10^8) \text{ rad/s}$
- Determine the permittivity ($\varepsilon$):
Using the given constant $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9$, we find the permittivity of free space: $\varepsilon_0 = \frac{1}{4\pi (9 \times 10^9)} = \frac{1}{36\pi \times 10^9} \text{ F/m}$. Assuming the relative permittivity $\varepsilon_r = 1$ (as is common when not specified), the medium's permittivity is $\varepsilon = \varepsilon_r \varepsilon_0 = \varepsilon_0$.
- Compute the ratio:
Ratio $R = \frac{\sigma}{\omega \varepsilon} = \frac{10 \text{ mho/m}}{(2\pi \times 10^8 \text{ rad/s}) \times (\frac{1}{36\pi \times 10^9} \text{ F/m})}$ $R = \frac{10 \times 36\pi \times 10^9}{2\pi \times 10^8}$ $R = \frac{10 \times 36 \times 10^9}{2 \times 10^8}$ $R = \frac{360 \times 10^9}{2 \times 10^8}$ $R = 180 \times 10^{(9-8)} = 180 \times 10 = 1800$.
Result
The ratio of maximum conduction current density to maximum displacement current density is $1800$.
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 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 ___________.
- The equation of the electric field of an electromagnetic wave propagating through free space is given by : $E = \sqrt{377} \sin(6.27 \times 10^3 t - 2.09 \times 10^{-5} x)\text{ N/C}$
The average power of the electromagnetic wave is $\left(\frac{1}{\alpha}\right)\text{ W/m}^2$. The value of $\alpha$ is ___________.
$\left[ \text{Take } \sqrt{\frac{\mu_0}{\epsilon_0}} = 377\text{ in SI units} \right]$ 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 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 ___________.
