C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College
Prof. Surajit Dhara SACT,
Dept. Of Physics, Narajole Raj College
C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits
❖ Kirchhoff’s Laws: Kirchhoff’s offered two laws which are applicable to complicated networks of conductor.
➢ Kirchhoff’s 1st Law: The algebraic sum of currents meeting at a junction point of conductors is always zero.
▪ Explanation: Let 𝑖1, 𝑖2, 𝑖3…… be the currents through conductors connected at a junction point. Then
𝑖1 + 𝑖2− 𝑖3+ 𝑖4 − 𝑖5 = 0
i.e. ∑ 𝒊 = 𝟎
▪ Sign Convension: The current towards the junction point = + Ve The current outwards the junction point = -Ve
C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College
▪ Proof : From the equation of continuity,
∇⃗⃗ . 𝑗 = −𝜕𝜌
𝜕𝑡
⇒ ∰ ∇⃗⃗ . 𝑗 𝑑𝑣 = − ∰𝜕𝜌
𝜕𝑡 𝑑𝑣
⇒ ∰ 𝑗 . 𝑛̂𝑑𝑠 = − ∰𝜕𝜌
𝜕𝑡 𝑑𝑣
⇒ ∑ 𝒊 = 𝟎 ; for steady current 𝜕𝜌
𝜕𝑡 = 0 N.B- 1. Kirchhoff’s 1st law being the law of conservation of electric charge.
2. It is known as Kirchhoff’s current law or KCL
➢ Kirchhoff’s 2nd Law: The algebraic sum of the product of current and resistance in each branch of a closed network of conductance is equal to the total emf in the circuit.
▪ Explanation : Let us take a closed loop ABCDEA, consisting branches having resistances 𝑟1, 𝑟2, 𝑟3, 𝑎𝑛𝑑 𝑒1, 𝑒2, 𝑒3. Then
∑ 𝒊𝒓 = ∑ 𝒆 N.B – 1. 2nd law being the law of conservation of energy.
2. It is called Kirchhoff’s voltage law or KVL.
C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College
❖ Production of Alternating Current: Let at any moment the magnetic flux linked up with the coil,
𝜑 = 𝑁 ∫ 𝐵⃗ . 𝑛̂𝑑𝑠
Where N is the total no. of turns of the coil, B is the magnetic flux density, 𝑛̂ is the unit vector normal to the plane of the coil.
If 𝜃 be the angle between 𝐵⃗ and 𝑛̂ , then
𝜑 = 𝐵𝐴𝑁𝑐𝑜𝑠𝜃 ….(1) If 𝜔 be the angular velocity then 𝜃 = 𝜔𝑡
.: Induced emf 𝑒 = −𝑑∅
𝑑𝑡
= − 𝑑
𝑑𝑡(𝐵𝐴𝑁𝑐𝑜𝑠𝜔𝑡)
= 𝐵𝐴𝑁𝜔𝑠𝑖𝑛𝜔𝑡 …(2) Where 𝐵𝐴𝑁𝜔 = 𝐸0, the peak value of induced emf.
Thus the current passing through R, 𝑖 = 𝐸
𝑅 =𝐸0
𝑅 𝑠𝑖𝑛𝜔𝑡
C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College
⇒ 𝑖 = 𝑖0𝑠𝑖𝑛𝜔𝑡 …..(3) Where 𝑖0 = 𝐸0
𝑅 the peak value of current.
❖ Average Value of Alternating emf / current over Complete Cycle : The instantaneous value of alternating emf
𝑬 = 𝑬𝟎𝑠𝑖𝑛𝜔𝑡 ….(1) Taking imaginary part Thus average value of alternating emf. over complete cycle ,
𝐸𝑎𝑣 = 1
𝑇∫ 𝐸𝑑𝑡
𝑇
0
= 1
𝑇∫ 𝐸0𝑇 0𝑠𝑖𝑛𝜔𝑡𝑑𝑡 = − 𝐸0
𝜔𝑇[𝑐𝑜𝑠𝜔𝑡]T0
= −𝐸0
2𝜋[𝑐𝑜𝑠2𝜋 − 𝑐𝑜𝑠0] 𝜔 = 2𝜋
𝑇 = 0
Similarly, 𝑖𝑎𝑣 = 0
❖ Average Value of Alternating emf / current over Half Cycle :
𝐸𝑎𝑣|𝑇
⁄2 = 1
𝑇⁄2∫ 𝐸𝑑𝑡
𝑇⁄2
0
= 2
𝑇∫0𝑇⁄2𝐸0𝑠𝑖𝑛𝜔𝑡𝑑𝑡 = −𝑖𝐸0
𝑤𝑇[𝑐𝑜𝑠𝑤𝑇
2 − 𝑐𝑜𝑠0]
= 2𝐸0
𝜋
Similarly, 𝑖𝑎𝑣|𝑇
⁄2 = 2𝑖0
𝜋
❖ Root mean square (R.M.S) value of Alternating emf/current:- The mean square value of alternating emf over a full cycle,
𝐸𝑚.𝑠 = 1
𝑇∫ 𝐸2𝑑𝑡
𝑇 0
C3T ( Electricity and Magnetism) , Topic :- Electrical Circuits: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College
= 1
𝑇∫ 𝐸2𝑠𝑖𝑛2𝜔𝑡𝑑𝑡
𝑇
0
= 𝐸02
2𝑇 ∫ (1 − 𝑐𝑜𝑠2𝜔𝑡)𝑑𝑡
𝑇
0
= 𝐸02 2
Thus root mean square (r.m.s) value of alternating emf, 𝑬𝒓.𝒎.𝒔 = 𝑬𝟎
√𝟐 Similarly, 𝒊𝒓.𝒎.𝒔 = 𝒊𝟎
√𝟐
➢ Form Factor : The ratio of r.m.s value of alternating emf. to the average value (over half cycle) is known as form factor.
.: Form factor F=𝐸𝑟.𝑚.𝑠
𝐸𝑎𝑣 = 𝜋
2√2 = 1.11
𝑭 = 𝟏. 𝟏𝟏
➢ Average Power calculation: Average power for a complete cycle of alternating currnt , average power,
𝑃𝑎𝑣 = 1
𝑇∫ 𝐸0𝑖𝑜𝑠𝑖𝑛2𝜔𝑡𝑑𝑡
𝑇 0
= 𝐸0𝑖𝑜 𝑇 ×𝑇
2 = 𝐸0
√2× 𝑖0
√2
.: 𝑷𝒂𝒗 = 𝑬𝒓.𝒎.𝒔× 𝒊𝒓.𝒎𝒔
