Revision: 12th Std >> Current Electricity MAH-MHT CET (PCM/PCB) Current Electricity

A current-measuring instrument which is always connected in series with a resistance RR through which the current is to be measured is called an ammeter.

The scalar product of current density vector and area vector, given by I = J ⋅ ΔS, which represents the flow of electric charge through a conductor, is called electric current.

The vector quantity J whose scalar product with the area vector ΔS gives the electric current through that area is called the current density vector.

An instrument used to measure the potential difference between two points in an electrical circuit, always connected in parallel with the component across which the voltage drop is to be measured, is called a voltmeter.

An arrangement of four resistors used to measure the resistance of one of them in terms of the other three, invented by Samuel Hunter Christie in 1833 and made famous by Sir Charles Wheatstone, is called a Wheatstone bridge.

The condition of the Wheatstone bridge under which the galvanometer shows zero (null) deflection, i.e., I_g = 0, is called the balance condition of the bridge.

A device, based on the Wheatstone bridge principle, which is used to measure the resistance of an unknown wire (conductor) with good accuracy is called a meter bridge (slide wire bridge).

An ideal apparatus of infinite resistance, based on the null deflection method, which is used to measure unknown potential differences accurately without drawing any current from the circuit, is called a potentiometer.

A potentiometer is a manually adjustable, variable resistor with three terminals. Two terminals are connected to the ends of a resistive element, and the third terminal is connected to an adjustable wiper. The position of the wiper sets the resistive divider ratio.

The potential gradient of a potentiometer wire is defined as the change in electric potential (voltage) per unit length of the wire.

Mathematically,

Potential Gradient = `V/L`

An electromechanical, sensitive instrument which is used to detect and measure small electric currents in a circuit is called a galvanometer.

Although represented with an arrow, current does not obey vector addition. It is a scalar given by:

I = J ⋅ ΔS

where J = current density vector and ΔS = area vector.

Balance condition (when I_g = 0):

Based on Wheatstone bridge principle:

R = S\[\left(\frac{l_1}{100-l_1}\right)\]

where R = unknown resistance, S = known resistance, l₁​ = distance of null point from the first end.

r = \[\left(\frac{l_1-l_2}{l_2}\right)\]R

\[\frac{E_1}{E_2}=\frac{l_1}{l_2}\]

\[\frac{E_1+E_2}{E_1-E_2}=\frac{l_1+l_2}{l_1-l_2}\]

S = \[\frac{G\cdot I_g}{I-I_g}\]

If current I = nI_g​: S = \[\frac {G}{n-1}\]

\[R_A=\frac{S\cdot G}{S+G}=\frac{G}{n}\]

At any junction, the sum of currents entering = the sum of currents leaving.

Example: I₁ + I₃ = I₂ + I₄​. Based on conservation of charge.

The algebraic sum of potential differences in a closed loop is zero.

Based on conservation of energy.

Statement

At any junction in an electric circuit, the sum of currents entering the junction is equal to the sum of currents leaving the junction. 

Derivation

When the current in a circuit is steady, charge does not accumulate at any junction. Therefore, the amount of charge entering the junction per second must be equal to the amount of charge leaving the junction per second. 

If currents I₁​ and I₂ enter a junction and currents I₃​ and I₄​ leave it, then

or

Hence,

Conclusion

Kirchhoff's First Law is a direct consequence of the conservation of charge. 

Statement

In any closed loop of an electric circuit, the algebraic sum of all changes in potential is zero. 

Derivation

Consider a charge moving around a closed loop. After completing one full loop, the charge returns to its starting point. Since electric potential depends only on position, the net change in potential over a complete loop must be zero. 

Therefore, in a closed loop,

If a loop contains cells and resistors, then the total emf supplied by the sources is equal to the total potential drop across the resistors. Thus,

Conclusion

Kirchhoff's Second Law is a direct consequence of the conservation of energy.

Let `I_3` and `I_4`  be the currents in resistors Q and S respectively . Let `I_g` be the current through galvanometer. For balanced condition, 

`I_g = 0`

Applying junction law at ‘b’ we get

`I_1 = I_3 + I_g`

`because I_g = 0 , I_1 = I_3`    ....(i)

Applying junction law at ‘d’, we get

`I_2 + I_g = I_4`

`because I_g = 0 , I_2 = I_4`    ....(ii)

Applying loop law in the loop abda, we get

`-I_1·P - I_g·Q + -I_2·R = 0`

⇒ `-I_1P + I_2R = 0`  (`because I_g = 0`)

⇒ `I_1P = I_2R`

⇒ `P/R = I_2/I_1`               ....(iii)

Applying loop law in the loop bcdb, we get

`-I_3·Q + I_4·S + I_g·6 = 0`

⇒ `-I_3·Q + I_4·S + 0 = 0  (because I_g =0)`

⇒ `-I_3Q = I_4S`

⇒ `Q/S = I_4/I_3`

⇒ `Q/S = I_2/I_1`             ...(iv) [using eq.(i) and (ii)]

From eq. (iii) and (iv), `P/ R = Q/s`

⇒ `P/Q = R/S`

This is the balanced condition. 

 V ∝ L ⇒ V = xL

• Kirchhoff's laws are used for complex circuits. 
• Kirchhoff's First Law: Total current entering a junction = total current leaving a junction. 
• Kirchhoff's Second Law: Total potential rise in a closed loop = total potential drop in the loop. 
• KCL is based on conservation of charge. 
• KVL is based on conservation of energy. 
• Mathematical forms are ∑I = 0 and ∑V = 0. 
• The correct sign convention is essential in numericals.