SATHEE: alternating-currents Question 10

alternating-currents Question 10

Question: Q. 1. Show that in the free oscillations of an $L C$ circuit, the sum of energies stored in the capacitor and the inductor is constant in time. U[CBSE SQP 2018-19]

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Solution:

Ans. At an instant $t$ , charge $q$ on the capacitor and the current $i$ are given by :

$\begin{aligned} q (t) = q_{0} \cos ω t & i (t) = - q_{0} ω \sin ω t \end{aligned}$

Energy stored in the capacitor at time $t$ is

$U_{K} = \frac{1}{2} C V^{2} = \frac{1}{2} \frac{q^{2}}{C} = \frac{q_{0}^{2}}{2 C} \cos^{2} (ω t) 1$

Energy stored in the inductor at time $t$ is

$\begin{aligned} U_{M} & = \frac{1}{2} L i^{2} & = \frac{1}{2} L q_{0}^{2} ω^{2} \sin^{2} (ω t) & = \frac{q_{0}^{2}}{2 C} \sin^{2} (ω t) (∵ ω = 1 / \sqrt{L C}) 1 \end{aligned}$

Sum of energies

$U_{E} + U_{M} = \frac{q_{0}^{2}}{2 C} (\cos^{2} ω t + \sin^{2} ω t)$

$= \frac{q_{0}^{2}}{2 C}$

This sum is constant in time as $q_{0}$ and $C$ , both are time-independent.

[CBSE Marking Scheme 2017]

[AT Q. 2. Obtain the expression for the energy density of magnetic field $B$ produced in the inductor.

U] [Delhi Comptt. 2016]

Ans. Instantaneous Induced $e m f$ in an inductor when current changes through it

$e = - L \frac{d I}{d t}$

Hence, instantaneous applied voltage

$e = V = L \frac{d I}{d t}$

Work done, $d W = V . d q =$ VIdt

$∴ d W = LIdI$

Long Answer Type Questions

$\begin{aligned} \Rightarrow \int d W & = \int_{0}^{I} L I d I W & = \frac{1}{2} L I^{2} \end{aligned}$

Energy density, $U = \frac{total energy stored}{volume} 1 / 2$

$U = \frac{(\frac{1}{2}) L I^{2}}{A l} = \frac{\frac{1}{2} (L I) I}{A l}$

$Flux = N B A = L I$

and

$B = \frac{μ_{0} N I}{l} \Rightarrow I = \frac{B l}{μ_{0} N} 1 / 2$

$∴ = \frac{\frac{1}{2} (N B A) \cdot \frac{B l}{μ_{0} N}}{A l} = \frac{B^{2}}{2 μ_{0}} 1 / 2$

[CBSE Marking Scheme 2016]

(5 marks each)

Alternating Currents

alternating-currents Question 10

Long Answer Type Questions

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Alternating Currents

alternating-currents Question 10

Long Answer Type Questions

Table of Contents

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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