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).

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`

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.

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.

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.

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