Overview: Current Electricity

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

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

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

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

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.

• 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.

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.

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

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 = ρ\[\frac {l}{A}\]

where:

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

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

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

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.

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

EMF Ratio (Sum & Difference Method):

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

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

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

• 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)