Concept:

Inductance:

• The induced emf across a coil is directly proportional to the rate of change of current through it. The proportionality constant in that relation is known as inductance.
• It is denoted by L and inductance of a coil is given by:

​

$L=\frac{\mu N^{2}A}{l}$

N = number of turns in the coil

A = Area of the cross-section

l = length of the coil

μ = magnetic permeability of the coil

Calculation:

Given that L₁ = 3 H, N₁ = 2000 and L₂ = 5 H

Inductance of a coil 

$L=\frac{\mu N^{2}A}{l}$

Now we can see that

L ∝ N²

$\begin{array}{l}\Rightarrow \frac{{\mathrm{L}}_1}{{\mathrm{L}}_2}=\frac{{\mathrm{N}}_1^2}{{\mathrm{N}}_2^2}\\ \Rightarrow {\mathrm{N}}_2^2={\mathrm{N}}_1^2\times \frac{{\mathrm{L}}_2}{{\mathrm{L}}_1}\\ \Rightarrow {\mathrm{N}}_2={\mathrm{N}}_1\sqrt{\frac{{\mathrm{L}}_2}{{\mathrm{L}}_2}}\\ \Rightarrow {\mathrm{N}}_2=2000\sqrt{\frac{5}{3}}=2582\end{array}$

Concept:

Parallel combination:

When two or more inductors are connected in such a way that their ends are connected at the same two points and have an equal potential difference for all inductors is called a parallel combination of the inductor.

Equivalent Inductance(L_eq) for parallel combination:

$\frac{1}{L_{eq}}=\frac{1}{L_1}+\frac{1}{L_2}+\frac{1}{L_3}$

Where

L₁ is the inductance of the first inductor,

L₂  is the inductance of the second inductor,

L₃  is the inductance of the third inductor,

Series combination:

When two or more inductors are connected end to end and have the same electric current on each is called the series combination of the inductors.​

Equivalent inductance (L_eq) in series combination:

L_eq = L₁ + L₂ + L₃

Calculation:

Inductance:

Inductance is the element property, in which energy is absorbed in the form of a magnetic field.

According to Faraday’s second law,

$\left|V\right|=N\frac{d\varphi }{dt}$                  ..... (1)

Also,

$N\varphi =Li$           ...... (2)

Where,

V = Voltage

ϕ = flux

N = number of turns

L = inductance

i = current

On comparing equation (1) & (2), we get

$V=\frac{d\left(Li\right)}{dt}$

$V=L\frac{di}{dt}$

Capacitance(C): 

• The capacity of a capacitor to store the electric charge is called capacitance.
• The capacitance of a conductor is the ratio of charge (Q) to it by a rise in its potential (V) 
• Capacitance ​C = Q/V
• The unit of capacitance is the farad or Coulomb/Volt  (symbol F ).
• Energy stored (U) in the capacitor is $U=\frac{1}{2}C{V}^2$

Q = charge in coulomb

C = Capacitance in farad

V = Voltage in volt

Combination of capacitors:

Parallel combination: 

When two or more capacitors are connected in such a way that their ends are connected at the same two points and have an equal potential difference for all capacitor is called a parallel combination of the capacitor.

Equivalent capacitance (Ceq) for parallel combination:

Ceq = C1 + C2 + C3

Where

C1 is the capacitance of the first capacitor,

C2 is the capacitance of the second capacitor

C3 is the capacitance of the third capacitor

Series combination:

When two or more capacitors are connected end to end and have the same electric charge on each is called the series combination of the capacitor.​

Equivalent capacitance (Ceq) in series combination:

$\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}$

Calculation:

Redrawing the circuit

Further simplifying

Step 1:

Both capacitor in series

Step 2:

Both capacitor in parallel

Step 3:

Both capacitor in series

Step 4:

Both capacitor in parallel

Equivalent capacitance

$C_{eq}=\frac{3}{5}C+C$

$C_{eq}=\frac{8C}{5}$

Concept:

Energy stored by a capacitor connected to a voltage source V is given by:

$E=\frac{1}{2}C{V}^2=\frac{1}{2}QV=\frac{1}{2}\frac{Q^2}{C}$

C = Capacitance of the capacitor

V = Voltage across the capacitor.

Q = charged stored

Application:

Given 'n' Capacitors are connected in parallel to a voltage source V.

So the voltage across each of the capacitors will be 'V' only.

Energy stored by this system of capacitors will be:

$E=\frac{1}{2}{C}_{net}{V}^2$

net capacitance of Capacitors are connected in parallel

C_net = C₁ + C₂ + C₃ + ... + C_n

Since all the capacitance is of equal value 'C'. The net capacitance will be:

$C_{net}=nC$. 

The energy stored is given as:

$E_{net}=\frac{1}{2}nC{V}^2$.

Concept:

Capacitance(C): 

• The capacity of a capacitor to store the electric charge is called capacitance.
• The capacitance of a conductor is the ratio of charge (Q) to it by a rise in its potential (V) 
• Capacitance ​C = Q/V
• The unit of capacitance is the farad, (symbol F ).

Q = charge in coulomb

C = Capacitance in farad

V = Voltage in volt

Calculation:

Given that, Q = 0.5 C, V = 20 V

The capacitance of the capacitor can be calculated as

C = Q/V

$C=\frac{0.5}{20}$

C = 0.025 F

Concept:

According to charge conservation, the capacitor with minimum charge(Q) will reach it’s saturation first and will breakdown first.

Charge (Q )of capacitor for given capacitance (C) & Voltage(V) is given by

Q = CV,​​

Combination of capacitors:

Parallel combination: 

When two or more capacitors are connected in such a way that their ends are connected at the same two points and have an equal potential difference for all capacitor is called a parallel combination of the capacitor.

Equivalent capacitance (C_eq) for parallel combination:

Ceq = C1 + C2 + C3

Where

C1 is the capacitance of the first capacitor,

C2 is the capacitance of the second capacitor

C3 is the capacitance of the third capacitor

Series combination:

When two or more capacitors are connected end to end and have the same electric charge on each is called the series combination of the capacitor.​

Equivalent capacitance (Ceq) in series combination:

$\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}$

Calculation:

Charge on each of the capacitor (Q) = 4500 μC

As all the capacitors are connected in series, the charge in the circuit is 4500 μC

Voltage across the series combination (V) = 135 V

Let the equivalent capacitance of series combination is C_eq

Q = C_eq V

4500 = C_eq × 135

C_eq = 33.33 μF

Concept-

• The coil which stores magnetic energy in a magnetic field is called as inductor.
• The property of an inductor which causes the emf to generate by a change in electric current is called as inductance of the inductor.
• The SI unit of inductance is Henry (H).

The magnetic potential energy stored in an inductor is given by:

$E=\frac{1}{2}L{I}^2$

$L=\frac{\mu N^{2}A}{l}$ = inductance of the inductor 

I = current

Calculation:

Energy stored in the inductor is given as:

$E=\frac{1}{2}L{i}^2$

The inductance of a coil is proportional to square of the number of turns

L ∝ N²

The stored energy if the number of turns is doubled and the current is halved

I₂ = I₁/2, L₂ = 4L₁

∴ Stored energy is

$E^{\prime }=\frac{1}{2}4L_1{\left(\frac{i_1}{2}\right)}^2=\frac{1}{2}{L}_1{i}_1^2=0.5J$

Concept:

The capacitance of a capacitor (C):

The capacitance of a conductor is the ratio of charge (Q) to it by a rise in its potential (V), i.e.

C = Q/V

The unit of capacitance is the farad, (symbol F).

Parallel Plate Capacitor:

A parallel plate capacitor consists of two large plane parallel conducting plates of area A and separated by a small distance d.

The formula for the capacitance of a parallel plate capacitor is given by

$C=\frac{A{ϵ}_0}{d}$

A = area of the plate

ϵ₀ = permittivity of vacuum = 8.85 × 10^-12

d = distance between plate 

Calculation:

Given that,

A = 0.01 m²

d = 2.5 cm = 0.025 m

ϵ₀ = 8.85 × 10-12 

Now  the capacitance of a parallel plate capacitor

$C=\frac{0.01\times 8.85\times {10}^{-12}}{0.025}$

C = 3.54 pF