Revision Notes on Electromagnetic Induction and Alternating Current

• Magnetic flux:- 

Magnetic flux lined with the surface is defined as the product of area and component of B perpendicular that area.

and

ϕ_B = µnAH

Here, µ is the permeability of the medium, n is the number of turns, A is the area and H is the magnetic field intensity.

(a) When θ = 90º, cosθ = 0. So, ϕ_B = 0

This signifies, no magnetic flux is linked with surface when the field is parallel to the surface.

(b) When θ = 0º, cosθ = 1. So,(ϕ_B)_max = 1

This signifies, magnetic flux linked with a surface is maximum when area is held perpendicular to the direction of field.

• Faraday’s law of electromagnetic induction:-

(a) Whenever magnetic flux linked with a circuit changes, an e.m.f is induced in it.

(b) The induced e.m.f exists in the circuit so long as the change in magntic flux linked with it continues.

(c) The induced e.m.f is directly proportional to the negative rate of change of magnetic flux linked with the circuit.

So, E = -dϕ_B/dt

Negative sign is due to the direction of induced e.m.f.

• Induced electric field:-

• Lenz’s Law:-

It states that direction of induced e.m.f. is such that it tends to oppose the very csause which produces it.

The induced e.m.f. always tends to oppose the cause of its production.

• Motion of a straight conductor in a uniform magnetic field:-

(a) W = Bevl

(b) Motional e.m.f, E = Bvl

(c) Induced current, I = E/R = Blv/R

(d) F = IlB = B²l²v/R

(e) P = Fv = IlBv = B²l²v²/R

(f) H = I²R = B²l²v²/R                             

• Motion of a loop in a magnetic field when whole of the coil is in the magnetic field:

(a) Motional e.m.f , E = 0

(b) Resultant Current, I = 0

(c) Force, F = 0

(d) Power, P = 0

• Motion of a loop in a magnetic field when a part of the loop is out of the magnetic field:-

(a) ϕ_B = Blx

(b) Induced e.m.f , E = Blv

• Power:-

P = I²R = E²/R

P = B²l²v²/R       (Since, E = Blv)

(a) Coil out of field:- ϕ_B =0, E = 0, P = 0

(b) Coil entering the magnetic field:-

ϕ_B increases gradually

E = a negative constant

P = a positive constant

(c) Coil moving in the magnetic field:-

ϕ_B = Constant

E = 0

P = 0

(d) Coil leaving the magnetic field:-

ϕ_B decreases gradually

E = a positive constant

P = a positive constant

(e) Coil out of magnetic field:-

ϕ_B = 0

E = 0

P = 0

• Self-Induction:- Self Induction of a circuit is defined as the property of the circuit by virtue of which it tends to oppose a change in the strength of current, through it, by inducing an e.m.f. in itself. 

(a) Magnetic flux, ϕ_B = LI

Here L is the coefficient of self -induction.

(b) e.m.f., E = -L [dI/dt]

(c) L = µ₀ µ_rnNA

Here, n is the number of turns per unit length

• Series and parallel combination:-

(a) L = L₁+L₂ (If inductors are kept far apart and joined in series)

(b) L = L₁+L₂±2M   (If inductors are connected in series and they have mutual inductance M)

(c) 1/L = 1/L₁ + 1/L₂ (If two conductors are connected in parallel and are kept for apart)

(d) M = K√L₁L₂      (If two coils of self-inductances, L₁ and L₂ are over each other)

• Inductance of wire:-

L = µ₀l/8π

• Inductance of hollow cylinder:-

L = µ₀l/2π [ln 2l/a -1],  l >> a

• Inductance of parallel wires:-

L = µ₀l/π [ln d/a -1], l >> d, d >> a

• Inductance of Coaxial conductor:-

L = µ₀l/π [ ln b/a]

• Inductance of Circular loop:-

L = µ₀l/2π [ln 4l/d – 2.45]

l = -2πρ₀, ρ₀ >> d

• Inductance of Solenoid:-

L = µ₀N²S/l

L >> a

• Inductance of Torus (of circular cross section):-

L = µ₀N² [ρ₀ - √ ρ₀² – a²]

• Inductance of Sheet:-

L = µ₀2l [ln (2l/b+t) + 0.5]

• Energy stored in an conductor:-

(a) W = ½ LI²

Here L is the coefficient of self -induction.

(b) U_B = B²/2µ₀

• Mutual Induction:-

Mutual induction of two circuits is the phenomenon where a current changing in the first coil results in the induction of an e.m.f. in the second.

• Coefficient of Mutual Induction:-

ϕ_B = MI and E = -M[dI/dt]

Here M is called the coefficient of mutual induction of two circuits.

The value of M, M =  µ₀ µ_rn₁ N₂ A

M depends upon,

(a) Area of cross-section of the two coils

(b) Number of turn of each coil

(c) Distance between the two coils

(d) Nature of material used as core

• Fleming’s right hand rule:-

Stretch first finger, central finger and the thumb of your right hand in three mutually perpendicular directions. If the first finger points towards the magnetic field, thumb points towards the direction of motion of conductor, the direction of central finger gives the direction of induced current set up in the conductor.

• Coil rotating in a uniform magnetic field:-

(a) Magnetic flux, ϕ_B = µnaH [cos ωt]

(b) Electromagnetic Induction, E = µnaωH [sin ωt]

(c) Current, I = [µnaωH[sin ωt]]/R

• Growth and decay of current in LR circuit:-

(a) I = I₀(1-e^-t/τ)         (for growth), Here τ = L/R

(b) I = I₀e^-t/τ               (for decay), Here τ = L/R

• Alternating Current:-

An alternating current (a.c.) is a current which continuously, changes in magnitude and periodically reverses in direction.`

i = I₀ sin ωt = I₀ sin (2π/T) t

Here I₀ is the peak value of a.c.

(a) Current, I =I₀ sin ωt

(b) Angular frequency, ω= 2πn   (n is the frequency of a.c.)

(c) I =I₀ sin 2πnt

• Mean value of A.C or D.C. value of A.C.:-

Mean value of a.c. is that value of steady current which sends the same amount of charge, through a circuit, in same time as is done by a.c. in one half-cycle.

(I_av)_{half cycle} = (2/π)I₀

Thus, mean value of alternating current is 2/π times (0.637 times) its peak value.

(V_av)_{half cycle} = (2/π) V₀

• Average value of A.C. over a complete cycle:-

I_av = 0

The average value of a.c. taken over the complete cycle of a.c.is zero.

• Root mean square value of a.c. or virtual value of a.c.:-

Root mean square value of alternating current is defined as that value of steady current which produces same heating effect, in a resistance, in a certain time as is produced by the alternating current in same resistance in same time. The r.m.s value of a.c.is also called its virtual value.

I_rms = I₀/√2

Root mean square value of alternating current is I/√2 times (or 0.707 times) the peak value of current.

Similarly, V_rms= V₀/√2

Here V₀ is the peak value of e.m.f.

• Form Factor:-

Form Factor = rms value/average value = (V₀/√2)/ (2 V₀/π)  = π/2√2

• Current elements:-

(a) Inductive reactance:- X_L = ωL

Here, ω = 2πn, n being frequency of a.c.

L is the coefficient of self-inductance of coil.

(b) Capacitative reactance:- X_c = 1/ωC

Here C is the capacity of the condenser

• Capacitor in AC circuit:-

q = CV₀sinωt

I = I₀ sin(ωt +π/2)

V₀ = I₀/ωC

X_c = 1/ωC

• Inductor in AC circuit:-

V_L = L(dI/dt) = LI₀ω cosωt

I = (V₀/ ωL) sinωt

Here, I₀ = V₀/ ωL

X_L = ωL

And the maximum current, I₀ = V₀/XL

• R-L circuit:-

I = ε/R [1-e^-Rt/L]

V = ε e^-Rt/L

• Graph between I (amp) and t (sec):-

• Graph between potential difference across inductor and time:-

• L-C Circuit:-

f = 1/2π√LC

q = q₀ sin (ωt+ϕ)

I = q₀ωsin (ωt+ϕ)

ω = 1/√LC

• The total energy of the system remains conserved,

  ½ CV² + ½ Li² = constant = ½ CV₀² = ½ Li₀²

• Series in C-R circuit:-

 V = IZ

The modulus of impedance, |Z |= √R²+(1/ωC)²

The potential difference lags the current by an angle, ϕ = tan^-1(1/ωCR)

• Series in L-C-R Circuit:

V = IZ

The modulus of impedance, |Z |= √[R²+(ωL-1/ωC)²]

The potential difference lags the current by an angle, ϕ = tan^-1[ωL -1/ωC)/R]

• Circuit elements with A.C:-

Circuit elements	Amplitude relation	Circuit quantity	Phase of V
Resistor	V0 = i0R	R	In phase with i
Capacitor	V0 = i0XC		Lags i by 90°
Inductor	V0 = i0XL	XL = wL	Leads i by 90°

• Resonance:-

(a) Resonance frequency:- f_r = 1/2π√LC

(b) At resonance, X_L = X_C, ϕ = 0, Z = R(minimum), cosϕ = 1, sinϕ = 0 nad current is maximum (=E₀/R)

• Half power frequencies:-

(a) Lower,  f₁ = f_r – R/4πL    or      ω₁ = ω_r – R/2L

(b) Upper,  f₂ = f_r + R/4πL    or      ω₂ = ω_r + R/2L

• Band width:- Δf = R/2πL   or   Δf = R/L

• Quality Factor:-

(a) Q = ω_r/Δω = ω_rL/R

(b) As ω = 1/√LC, So Q ∝ √L, Q ∝1/R and Q ∝ 1/√C

(c) Q = 1/ω_rCR

(d) Q = X_L/R   or  Q = X_C/R 

(e) Q = f_r/Δf   

• At resonance, peak voltages are:-

(a) (V_L)_res = e₀Q

(b) (V_C)_res = e₀Q

(c) (V_R)_res = e₀

• Conductance, susceptance and admittance:-

(a) Conductance, G = 1/R

(b) Susceptance, S = 1/X

(c) S_L = 1/X_L and S_C = 1/X_C = ωC

(d) Admittance, Y = 1/Z

(e) Impedance add in series while add in parallel

• Power in AC circuits:-

Circuit containing pure resistance:- P_av = (E₀/√2)×(I₀/√2) = E_v×I_v

Here E_v and I_v are the virtual values of e.m.f and the current respectively.

Circuit containing impedance (a combination of R,L and C):-

P_av = (E₀/√2)×(I₀/√2) cosϕ = (E_v×I_v) cosϕ

Here cosϕ is the power factor.

(a) Circuit containing pure resistance, P_av = E_vI_v

(b) Circuit containing pure inductance, P_av = 0

(c) Circuit containing pure capacitance, P_av = 0

(d) Circuit containing resistance and inductance,

Z = √R²+(ωL)

cosϕ = R/Z = R/[√{R²+(ωL)²}]

(e) Circuit containing resistance and capacitance:-

Z = √R²+(1/ωC)²

cosϕ = R/Z = R/[√{R²+(1/ωC)²}]

(f) Power factor, cosϕ = Real power/Virtual power = P_av/E_rmsI_rms

• Transformer:-

(a) C_p = N_p (dϕ/dt) and e_s = N_s (dϕ/dt)

(b) e_p/e_s = N_p/N_s

(c) As, e_pI_p = e_sI_s, Thus, I_s/I_p = e_p/e_s = N_p/N_s

(d) Step down:- e_s < e_p, N_s< N_p and I_s> I_p

(e) Step up:- e_s >e_p, N_s>N_p and I_s< I_p

(f) Efficiency, η = e_s I_s/ e_p I_p

• AC Generator:-

e = e₀ sin (2πft)

Here, e₀ = NBAω