Revision: Current Electricity Physics HSC Science (General) 12th Standard Board Exam Maharashtra State Board

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.

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 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.

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

The current sensitivity of a galvanometer is defined as the deflection produced in the galvanometer when a unit current flows through it.  
Mathematically, it can be given by:

I_S = `(NBA)/k`

Where k is the couple per unit twist.

Current sensitivity is defined as the deflection e per unit current.

A Moving Coil Galvanometer (MCG) is a sensitive electromagnetic instrument used to detect and measure small electric currents (of the order of microamperes to milliamperes) by measuring the deflection of a current-carrying coil placed in a uniform magnetic field.

Deflection produced per unit current.

Deflection produced per unit voltage.

The current required to produce a unit deflection (1 division) on the scale.

A galvanometer is a sensitive instrument used to detect and measure small electric currents in a circuit.

An ideal voltmeter is one which has infinite resistance and does not draw any current from the circuit.

Any closed conducting path in an electric network is called a loop or mesh.

Any point in an electric circuit where two or more conductors are joined together is a junction.

A branch is any part of the network that lies between two junctions.

A voltmeter is an instrument used to measure the potential difference between two points in an electric circuit.

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):

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

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

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

VS = \[\frac{\phi}{V}=\frac{NAB}{CG}\]

where G = resistance of the galvanometer coil.

Unit: div/V

CS = \[\frac{\phi}{I}=\frac{NAB}{C}\]

Unit: div/A or div/μA

k = \[\frac{I}{\phi}=\frac{C}{NAB}\]

k is the reciprocal of current sensitivity. A galvanometer with a smaller figure of merit is more sensitive.

R = ρ\[\frac {l}{A}\]

where:

• ρ = resistivity (specific resistance)
• l = length of wire
• A = cross-sectional area

\[\frac {X}{R}\] = \[\frac {l_1}{l_2}\]

or

X = \[R\frac{l_1}{100-l_1}\]

where:

• X = unknown resistance
• R = known resistance (from resistance box)
• l₁ = length of wire from one end to the balance point
• l₂ = remaining length of wire
• Total length = 100 cm

r = R\[\left(\frac{l_1}{l_2}-1\right)\]

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

EMF Ratio (Sum & Difference Method):

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

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

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

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. 

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. 

Step 1: Torque due to current (Deflecting Couple):

• When current I flows through a coil of N turns, area A, in a field B: τ_deflecting = N I A B (Since radial field: sin⁡90° = 1)

Step 2: Restoring Torque (Spring):

• The phosphor-bronze strip/spring opposes the deflection. If ϕ is the angular deflection and C (or k) is the torsional constant of the spring, τ_restoring = Cϕ

Step 3: Equilibrium Condition:

• At equilibrium, deflecting torque = restoring torque: NIAB = Cϕ

Step 4: Current–Deflection Relationship:

Statement

The potential difference between two points of a uniform wire carrying a constant current is directly proportional to the length of the wire between those points.

V ∝ l or V = Kl

where K is the potential gradient.

Explanation / Proof

When a steady current flows through a uniform wire,

V = IR

Since the resistance of the wire,

R ∝ l

Therefore,

V ∝ l

Thus, potential difference is directly proportional to the length of the wire.

Conclusion

Hence, proved that in a potentiometer, the potential difference varies directly with length, provided the current and temperature remain constant.

Statement

The algebraic sum of all potential differences (voltage drops) and electromotive forces (emfs) in any closed loop of an electrical circuit is zero.

∑I R + ∑ ε = 0

This means that the total voltage supplied in a closed loop is equal to the total voltage drop in that loop.

Sign Convention

• Across a Resistor:
  If the loop is traced in the direction of current, the potential drop (IR) is taken as negative.
  If the loop is traced against the direction of current, the potential drop (IR) is taken as positive.
• Across a Source (emf):
  Moving from negative to positive terminal inside the source → emf is taken as positive.
  Moving from positive to negative terminal inside the source → emf is taken as negative.

Statement

The algebraic sum of currents at any junction in an electrical network is zero.

∑I = 0

This means that the total current entering a junction equals the total current leaving it.

Sign Convention

• Currents entering the junction are taken as positive.
• Currents leaving the junction are taken as negative.

Thus,

I₁ + I₃ + I₄ − I₂ − I₅ − I₆ = 0

or

I₁ + I₃ + I₄ = I₂ + I₅ + I₆​

Statement

A Wheatstone bridge is said to be balanced when no current flows through the galvanometer.
Under this condition, the ratio of resistances in one pair of opposite arms is equal to the ratio in the other pair.

\[\frac {P}{Q}\] = \[\frac {S}{R}\]​

Proof / Explanation

Consider four resistances P, Q, R, S forming a bridge.

When the bridge is balanced:

I_g = 0

(No current flows through the galvanometer.)

Applying Kirchhoff’s Voltage Law to the loops:

From loop 1:

I₁P = I₂S

From loop 2:

I₁Q = I₂R

Dividing the two equations:

\[\frac {P}{Q}\] = \[\frac {S}{R}\]​

Conclusion

When the bridge is balanced:

• No current flows through the galvanometer.
• The above ratio condition holds.
• If any three resistances are known, the fourth can be determined.

• 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. 

• Choose some direction of the currents.
• Reduce the number of variables using Kirchhoff's first law.
• Determine the number of independent loops.
• Apply voltage law to all the independent loops.
• Solve the equations obtained simultaneously.
• In case, the answer of a current variable is negative, the conventional current is flowing in the direction opposite to that chosen by us.

• The Wheatstone bridge is used for measuring the values of very low resistance precisely.
• We can also measure quantities such as galvanometer resistance, capacitance, inductance and impedance using a Wheatstone bridge.

• A potentiometer can be used as a voltage divider, where the output voltage is proportional to the length of the wire segment:
  V ∝ l
• It is used in audio control systems (sliders and rotary knobs) for loudness control and frequency adjustment.
• A potentiometer can work as a motion/displacement sensor, where change in position produces proportional change in potential difference.
• It is more sensitive and more accurate than a voltmeter, capable of measuring very small potential differences (of the order of 10⁻⁶ V).
• A potentiometer can measure both emf and potential difference, whereas a voltmeter measures only terminal potential difference.
• Limitations: It is not portable and does not provide direct reading; balancing (null point) is required for measurement.

• A moving-coil galvanometer (MCG) can be converted into an ammeter by connecting a low-resistance shunt (S) in parallel with the galvanometer to increase its current range.
• An ideal ammeter should have zero resistance; the shunt decreases the effective resistance and protects the galvanometer from excess current.
• If G is the galvanometer resistance, I_{g is the} full-scale deflection current, and I is the total current, then shunt resistance:
  S = \[\frac {GI_g}{I-I_{g}}\]​​
• To increase the range nnn times (I = nI_g):
  S = \[\frac {G}{n-1}\]​

• A moving-coil galvanometer (MCG) can be converted into a voltmeter by connecting a high resistance (X) in series to increase its voltage range.
• An ideal voltmeter should have very high (ideally infinite) resistance and always be connected in parallel across the component.
• If G is the galvanometer resistance and I_g is full-scale deflection current, then the required series resistance:
  X = \[\frac {V}{I_g}\] - G
• If the voltage range is increased nnn times, then:
  X = G(n − 1)