JEE Physics Current Electricity Notes Set 05

Cells, EMF and Internal Resistance

A cell converts chemical energy into electrical energy and maintains a potential difference between its terminals. Its emf \( \epsilon \) is the terminal voltage when no current is drawn. When a current I flows, part of the emf is lost across the cell's own resistance r, so the terminal voltage is \( V=\epsilon-Ir \) while discharging and \( V=\epsilon+Ir \) while charging.
For an external resistance R, the current is \( I=\frac{\epsilon}{R+r} \), and the internal resistance can be found from \( r=\left(\frac{\epsilon}{V}-1\right)R \).
Maximum power is delivered to the load when \( R=r \); the maximum power is then \( \frac{\epsilon^2}{4r} \) and the efficiency is only 50%.

Grouping of Cells

n identical cells in series give emf \( n\epsilon \) and internal resistance nr. n identical cells in parallel give emf \( \epsilon \) and internal resistance r/n. In a mixed grouping of m rows with n cells in each row, the emf is \( n\epsilon \) and the internal resistance is \( \frac{nr}{m} \); the current is largest when \( R=\frac{nr}{m} \).
Two unequal cells in parallel (like poles together) behave as a single cell with \( \epsilon_{eq}=\frac{\epsilon_1r_2+\epsilon_2r_1}{r_1+r_2} \) and \( r_{eq}=\frac{r_1r_2}{r_1+r_2} \), so the equivalent emf lies between the two emfs.

Kirchhoff's Rules and Circuit Analysis

Junction rule: the algebraic sum of currents at a junction is zero, which expresses conservation of charge. Loop rule: the algebraic sum of potential changes around any closed loop is zero, which expresses conservation of energy.
Sign convention: crossing a resistor in the direction of current is a fall of iR; crossing a cell from its negative to its positive terminal is a rise equal to its emf.
Shortcuts: points at the same potential can be joined; a balanced Wheatstone bridge carries no current in its middle arm; by symmetry, equal currents flow in equivalent branches (for example, the edges of a cube give 5R/6 between opposite corners).

Wheatstone Bridge and Metre Bridge

A Wheatstone bridge is balanced when \( \frac{P}{Q}=\frac{R}{S} \). In a metre bridge the ratio of the two gap resistances equals the ratio of the wire lengths on either side of the null point: \( \frac{X}{R}=\frac{l}{100-l} \).
Interchanging the two gap resistances inverts the ratio and is used to remove end-correction errors. The best accuracy is obtained when the null point lies near the middle of the wire.

Potentiometer

A potentiometer is a long uniform wire carrying a steady current from a driver cell, giving a uniform potential gradient \( k=\frac{V_{wire}}{L}=\frac{IR_{wire}}{L} \). It draws no current from the cell under test at the null point, so it measures the true emf.
Emf comparison: \( \frac{\epsilon_1}{\epsilon_2}=\frac{l_1}{l_2} \). Internal resistance: if the balance length is \( l_1 \) on open circuit and \( l_2 \) with a shunt resistance R across the cell, then \( r=R\left(\frac{l_1-l_2}{l_2}\right) \).
Sensitivity increases when the potential gradient is reduced, by adding a series resistance to the driver circuit.

Galvanometer, Ammeter and Voltmeter

A galvanometer of resistance G and full-scale current \( I_g \) becomes an ammeter of range I by a shunt \( S=\frac{I_gG}{I-I_g} \) in parallel, and a voltmeter of range V by a series resistance \( R=\frac{V}{I_g}-G \).
An ideal ammeter has zero resistance and an ideal voltmeter has infinite resistance. A real voltmeter reads less than the true voltage across a resistance because it draws some current; a real ammeter reads less than the true current because of its own resistance.

Resistivity and Temperature Effects

For a wire of fixed volume, resistance is proportional to the square of the length. In series, \( \rho_{eq}=\frac{\rho_1l_1+\rho_2l_2}{l_1+l_2} \) for wires of the same area. For a metal, \( R_t=R_0(1+\alpha t) \); when the length expands, the apparent coefficient of resistance is \( \alpha_\rho-\alpha_l \).
Two resistors in series keep a constant total resistance when \( R_1\alpha_1+R_2\alpha_2=0 \), which needs one positive and one negative coefficient.

Heating Effect of Current

Joule's law gives heat \( H=I^2Rt=\frac{V^2}{R}t=VIt \). In series, heat is proportional to resistance; in parallel it is inversely proportional to resistance. A bulb rated \( (P_0, V_0) \) has resistance \( \frac{V_0^2}{P_0} \) and draws power \( P=P_0\left(\frac{V}{V_0}\right)^2 \) at voltage V if its resistance is taken as constant.
Of two bulbs in series the one with the higher resistance (lower rating) is brighter; in parallel the one with the lower resistance (higher rating) is brighter. A fuse wire melts at a current that depends on its material and radius but not on its length.

Thermoelectric and Chemical Effects

Seebeck effect: a temperature difference between the junctions of two different metals produces an emf. For \( E=at+bt^2 \), the thermoelectric power is \( \frac{dE}{dt}=a+2bt \), the neutral temperature is \( t_n=-\frac{a}{2b} \), and the temperature of inversion is \( t_i=2t_n-t_0 \). The neutral temperature depends only on the metals. The Peltier coefficient is \( \pi=T\frac{dE}{dT} \) and the Thomson heat absorbed is \( \sigma\,\Delta T\,q \).
Faraday's laws of electrolysis: \( m=Zq=Zit \), and for the same charge through different cells \( \frac{m_1}{m_2}=\frac{E_1}{E_2} \), where E is the equivalent weight (atomic weight divided by valency). One faraday, 96500 C, deposits one gram-equivalent of a substance.