Resistance Calculation from Power Dissipation
The thermal energy dissipated ($P_R$) in an inductor due to its resistance ($R$) is given by the formula $P_R = I_{rms}^2 R$, where $I_{rms}$ is the root-mean-square current.
- Given $P_R = 32\text{ W}$ and $I_{rms} = 2\text{ A}$.
- Substitute the values: $32\text{ W} = (2\text{ A})^2 \times R$.
- $32 = 4 \times R$.
- Solving for $R$: $R = \frac{32}{4} = 8\text{ }\Omega$.
Inductance Calculation from Average Energy Stored
The energy ($U$) stored in an inductor's magnetic field varies with current. The problem states $16\text{ J}$ is stored. This is interpreted as the average energy stored ($\langle U \rangle$) over an AC cycle.
The average energy stored in an inductor is related to the peak energy ($U_{peak}$) by $\langle U \rangle = \frac{1}{2} U_{peak}$.
The peak energy is given by $U_{peak} = \frac{1}{2} L I_{peak}^2$, where $L$ is the inductance and $I_{peak}$ is the peak current.
First, find the peak current ($I_{peak}$) from the RMS current ($I_{rms}$): $I_{peak} = I_{rms} \sqrt{2} = 2\sqrt{2}\text{ A}$.
Now, relate average energy to inductance:
- $\langle U \rangle = \frac{1}{2} \left( \frac{1}{2} L I_{peak}^2 \right) = \frac{1}{4} L I_{peak}^2$.
- Given $\langle U \rangle = 16\text{ J}$.
- Substitute values: $16\text{ J} = \frac{1}{4} L (2\sqrt{2}\text{ A})^2$.
- $16 = \frac{1}{4} L (4 \times 2) = \frac{1}{4} L (8) = 2L$.
- Solving for $L$: $L = \frac{16}{2} = 8\text{ H}$.
Inductive Reactance Calculation
The inductive reactance ($X_L$) is calculated using the formula $X_L = 2\pi f L$, where $f$ is the frequency and $L$ is the inductance.
- Given $f = 50\text{ Hz}$ and $L = 8\text{ H}$. Use $\pi = 3.14$.
- $X_L = 2 \times 3.14 \times 50\text{ Hz} \times 8\text{ H}$.
- $X_L = 6.28 \times 400$.
- $X_L = 2512\text{ }\Omega$.
Ratio Calculation ($X_L / R$)
Finally, calculate the ratio of inductive reactance to resistance.
- $X_L = 2512\text{ }\Omega$.
- $R = 8\text{ }\Omega$.
- Ratio = $\frac{X_L}{R} = \frac{2512\text{ }\Omega}{8\text{ }\Omega}$.
- Ratio = $314$.
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 __________.
$\left[ \text{Take } \frac{1}{4\pi\varepsilon_0} = 9 \times 10^9\text{ Nm}^2\text{/C}^2 \right]$ - 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 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 __________.
$\left[ \text{Take } \frac{1}{4\pi\varepsilon_0} = 9 \times 10^9\text{ Nm}^2\text{/C}^2 \right]$ - 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 ___________.
