Overview: DC Circuits and Measurements

An electric cell is a source of electrical energy which maintains a continuous flow of charge in a circuit.

The work done by the cell in forcing a unit positive charge to flow in the whole circuit (including the cell) is called the ‘electromotive force' (emf) of the cell.

Mathematically,
E = \[\frac {dW}{dq}\]

OR

The potential difference between the positive and negative terminals of a cell in an open circuit (when no current flows).

If in the flow of 1 C of charge in a circuit, the work done by the cell be 1 J, then the emf of the cell is 1 V.

The terminal potential difference of a cell is equal to the work done for the flow of a unit charge in the external circuit only.

Mathematically,
V = \[\frac {W_{ext}}{q}\]

Metre bridge is a sensitive device based on the principle of Wheatstone's bridge, for the determination of the resistance of a conductor (wire).

It is an important instrument for measuring the emf of a cell or the potential difference between two points of an electric circuit.

r = R\[\left[\frac{E}{V}-1\right]\]

Cells in Series:

r_eq​ = r₁​ + r₂​ + …

I = \[\frac{nE}{nr+R}\]

\[\frac{1}{r_{eq}}=\frac{1}{r_1}+\frac{1}{r_2}+\ldots\]

I = \[\frac{E}{\left(\frac{r}{n}+R\right)}=\frac{nE}{r+nR}\]

I = \[\frac{mnE}{nr+mR}\]

Statement

In an electric circuit, the 'algebraic' sum of the currents meeting at any junction in the circuit is zero, that is, ∑ I = 0.

Proof

When applying this law, currents entering the junction are taken as positive, while currents leaving the junction are taken as negative.

Consider a junction O where five conductors meet, carrying currents I₁, I₂, I₃, I_4​ and I₅.
Let I₁ and I₂ enter the junction, and I₃, I_4​ and I_5​ leave the junction.

According to Kirchhoff’s first law,

∑ I = 0

That is,

I₁ + I₂ − I₃ − I₄ − I₅ = 0

or,

I₁ + I₂ = I₃ + I₄ + I₅​

Thus, the sum of incoming currents is equal to the sum of outgoing currents.

Conclusion

Kirchhoff’s first law, also called Kirchhoff’s current law (KCL), states that when a steady current flows in a circuit, no charge accumulates at any junction. Hence, the law is a direct consequence of the principle of conservation of electric charge.

Statement

In any closed loop of a circuit, the algebraic sum of the products of current and the resistance in each part of the loop is equal to the algebraic sum of the emfs in that loop, that is,
∑ IR = ∑ E

Proof

While applying Kirchhoff’s second law, the following sign conventions are used:

1. When we traverse a resistance in the direction of current, the product I R is taken as positive.
2. The emf is taken as positive when we traverse from the negative to the positive electrode of the cell through the electrolyte.

Consider the circuit shown, containing two cells of emfs E₁​ and E₂​ and three resistances R₁, R_2​, and R₃​. Let the currents in R_1​ and R₂ be I₁​ and I₂, respectively. Applying Kirchhoff’s first law at junction A, the current through R₃​ is I₁ + I₂.

The circuit has two closed loops.

For loop 1, applying Kirchhoff’s second law:

I₁R₁ − I₂R₂ = E₁ − E₂​

For loop 2, applying Kirchhoff’s second law:

I₂R₂ + (I₁ + I₂)R₃ = E₂

Conclusion​

Kirchhoff’s second law, also called Kirchhoff’s Voltage Law (KVL), is a direct statement of the conservation of energy. Using the equations obtained from this law, the currents in different parts of the circuit can be determined.

Statement

When a Wheatstone bridge is balanced, that is, when there is no deflection in the galvanometer, the ratio of the resistances of any two adjacent arms is equal to the ratio of the resistances of the remaining two adjacent arms, i.e.,
\[\frac {P}{Q}\] = \[\frac {R}{S}\]

Explanation/Proof

In a Wheatstone bridge, four resistances P, Q, R, and S are connected to form the four arms of a parallelogram. A galvanometer is connected across one diagonal and a cell across the other diagonal.

When the key is pressed, the current entering the junction is divided into two parts: current I₁​ flows through arm AB, and current I₂​ flows through arm AD. The resistances are adjusted such that there is no current through the galvanometer, indicating that the bridge is balanced.

Since there is no current in the diagonal BD, the same current I₁ flows through arms AB and BC, and the same current I₂​ flows through arms AD and DC.

Applying Kirchhoff’s second law to loop ABDA,

I₁P − I₂R = 0 ⇒ I₁P = I₂R   ---(i)

Applying Kirchhoff’s second law to loop BCDB,

I₁Q − I₂S = 0 ⇒ I₁Q = I₂S   ---(ii)

Dividing equation (i) by equation (ii),

\[\frac {P}{Q}\] = \[\frac {R}{S}\]

Conclusion​

Thus, when a Wheatstone bridge is balanced, the ratio of the resistances in one pair of adjacent arms is equal to the ratio of the resistances in the other pair of adjacent arms. This principle is used to determine an unknown resistance accurately.

• Battery and the need for a combination of cells
  A single cell cannot give a strong current, so two or more cells are combined to form a battery to obtain a suitable current or emf.
• A series combination of cells
  In a series combination, emfs and internal resistances add up. It is useful when the external resistance is much larger than the internal resistance.
• Parallel combination of cells
  In parallel combination, the emf remains the same as one cell, but the internal resistance decreases. It is useful when the external resistance is small.
• Mixed grouping of cells
  In mixed grouping, cells are connected in series and parallel to obtain both a large current and a suitable emf.
  I_max = \[\frac {nE}{2R}\]
• Condition for maximum current in mixed grouping
  Maximum current flows when the battery's internal resistance equals the external resistance. i.e. \[\frac {nr}{m}\] = R

• Purpose of a rheostat: A rheostat is used to control the current in an electric circuit by changing resistance.
• Construction: It has a Nichrome wire wound on a china-clay cylinder with a sliding contact.
• As a current controller: When connected through A–C or B–C, moving the sliding contact changes the current in the circuit.
• As a potential divider: When connected across A and B, and the circuit is taken from A–C (or B–C), the rheostat provides a variable fraction of the applied potential difference.
• Working principle: Sliding the contact changes the wire's effective length, thereby changing its resistance.

• Principle: The metre bridge works on the Wheatstone bridge principle, and balance is obtained at the null point where the galvanometer shows no deflection.
• Null point condition: At the null point, points B and D are at the same potential and
  \[\frac {P}{Q}\] = \[\frac {R}{S}\]
• Finding unknown resistance: If the wire is divided into lengths l and 100 − l, the unknown resistance is
  S = R\[\frac {(100−l)}{l}\].
• Reducing errors: Errors are reduced by interchanging the known and unknown resistances and taking the mean value.
• Precautions: Keep the null point near the middle, avoid heating the wire, and press the jockey lightly without rubbing.

• Null-deflection method: At balance, no current flows through the galvanometer, making the measurement independent of the cell's internal resistance.
• Uniform wire requirement: The potentiometer wire must have a uniform cross-section and material so that the potential drop along the wire is uniform.
• True emf measurement: The emf is measured in open circuit, ensuring the true value of the emf is obtained without energy loss in the cell.
• Sensitivity dependence: The sensitivity of a potentiometer increases as the potential gradient decreases, using a long wire and low current.
• Experimental precautions: Current should not flow for a long time to avoid heating of the wire, and touch the jockey lightly to prevent wire damage.