Cells, EMF, & Internal Resistance

Battery and EMF

• Battery: A battery is a device that maintains a potential difference between its two terminals. It supplies electrical energy to a circuit by converting chemical (or other) energy into electrical energy.
• Cell: A single electrochemical element that produces an emf by chemical reactions is called a cell. A battery may be a single cell or a combination of cells.
• Inside a battery there is an internal mechanism (chemical reactions, in a chemical cell) that exerts non-electrostatic forces on charges in the battery material. These forces drive positive charges toward one terminal and negative charges toward the other terminal.

We represent the force on a positive charge q due to the battery mechanism as follows:

As positive charge accumulates on terminal A and negative charge on terminal B, a potential difference develops between A and B. This accumulation creates an electrostatic field inside the battery which exerts an electrostatic force on charges

The electrostatic force on a positive charge q is opposite in direction to the battery (non-electrostatic) force. In steady state, the charge accumulation is such that the non-electrostatic battery force is balanced by the electrostatic force and no further net accumulation takes place

• If a charge q is taken from terminal B to terminal A, the work done by the battery force Fb over the path of length d is W = Fb d, where d is the separation between A and B.
• The work done by the battery force per unit charge is W/q = Fb d / q. This quantity is called the electromotive force (emf) of the battery. Although called a force, emf is actually energy (work) per unit charge; it has units of volt (J C-1).
• When the terminals are not externally connected, the electrostatic force Fe inside balances Fb; Fe = Fb = qE, where E is the internal electrostatic field. Then Fb d = q E d = qV, where V is the potential difference between the terminals. Thus the emf E of the battery equals the terminal potential difference when no external circuit is connected.
• Important distinction: emf is the work done per unit charge by the non-electrostatic battery mechanism; potential difference originates from the electrostatic field produced by charges on the terminals. Their magnitudes may be equal in open-circuit conditions, but they are conceptually different.
• Typical chemical cells are made by placing two different metals (or electrodes) in an electrolyte. Chemical reactions at the electrodes produce the emf.

• When the terminals A and B are connected by a conducting wire, current flows because there is an electric field in the wire that drives free charges. Electrons in the wire move opposite to the electric field; conventional current is in the direction of the field.
• Electrons arriving at terminal A from the wire would reduce the terminal's positive charge and hence reduce the terminal potential difference. The internal battery mechanism continues to push charges so as to restore the terminal potential difference; thus a steady current is maintained.
• An equivalent simplified model treats the battery as supplying positive charge into the external circuit; the external electric field pushes these charges through the circuit and they return to the battery's negative terminal. The battery mechanism then does work to move the charges back against the internal field, completing the cycle.
• Charging and discharging: If current is driven into the battery in the reverse direction (by an external source), charges are moved opposite to the battery's usual discharge direction and chemical reactions reverse; this is charging. When the battery delivers current to the external circuit, it is discharging.

Examples - Terminal potentials and currents

Example 1.

Find v_A - v_B

Sol.

v_A - iR - E = v_B

Rearranging, v_A - v_B = iR + E

Substituting the given values, v_A - v_B = 4 + 10 = 14 volt

Example 2. Shown in the figure. Find the current in the wire BD

Sol.

Let the potential at point D be 0 V and determine the potentials at other points from the emf and resistances.

Current in wire AD =

= 5 A from A to D

Current in wire CB =

= 4 A from C to B

Therefore, current in wire BD = 1 A from D to B.

Example 3. Find the current in each wire

Sol.

Let potential at point A be 0 V and compute potentials at other nodes as shown in the figure.

Current in BG = 40-0/1

= 40 A from G to B

Current in FC =

= 15 A from C to K

Current in DE from D to E

Current in wire AH = 40 - 35 = 5 A from A to H

Combinations of Cells

• A single cell often cannot provide the desired voltage or current. Two or more cells are therefore combined to obtain larger emf or higher current capability.
• Cells can be combined in three ways: (i) in series, (ii) in parallel, and (iii) in mixed grouping (series-parallel combinations).

Cells in Series

• In series combination, the negative terminal of each cell is connected to the positive terminal of the next cell. If n identical cells, each of emf E and internal resistance r, are connected in series then:
• Net emf of the series combination = nE.
• Total internal resistance = n r.
• Total resistance of the circuit when connected to an external resistance R = n r + R.

If the total current in the circuit is I, then

Case (i): If n r ≪ R, then

i.e., if total internal resistance is much less than the external resistance, the current from the series combination is approximately n times the current from a single cell. Thus cells with small internal resistance are usefully joined in series to obtain a larger current.

Case (ii): If n r ≫ R, then the total internal resistance dominates and the current from n cells in series is nearly the same as from a single cell. In this case series connection gives little advantage.

Cells in Parallel

When all cells have equal emf and internal resistance

• In parallel combination, all positive terminals are connected together and all negative terminals are connected together. If n identical cells with emf E and internal resistance r are connected in parallel across external resistance R then:
• Net emf of the parallel combination = E (same as one cell).
• Total internal resistance = r / n.
• Total resistance of the circuit = (r / n) + R.
• If the current in the external resistance is I then

Case I: If r ≪ R, then

i.e., if each cell's internal resistance is much less than the external resistance, the total current is approximately equal to the current from a single cell; parallel connection gives little improvement.

Case II: If r ≫ R, then

i.e., if each cell has a high internal resistance compared with the external resistance, the total current from n cells in parallel is nearly n times the current of one cell. Thus cells with high internal resistance should be connected in parallel to increase usable current.

When emf's and internal resistances of parallel cells differ

For non-identical cells connected in parallel across external resistance R, the total current is obtained by applying Kirchhoff's laws. Consider three cells with emf E₁, E₂, E₃ and internal resistances r₁, r₂, r₃, giving branch currents i₁, i₂, i₃. Kirchhoff's current law gives:

I = i₁ + i₂ + i₃

Applying Kirchhoff's voltage law to each branch yields expressions of the form:

IR + i₁ r₁ = E₁ 

Similarly for the other branches:

Substituting these into I = i₁ + i₂ + i₃ gives:

If n cells are joined in parallel, an equivalent emf E_eq and internal resistance r_eq can be defined by:

Cells in Mixed Grouping (Series-Parallel)

In mixed grouping some rows contain cells joined in series and several such rows are then connected in parallel. Suppose each cell has emf E and internal resistance r. If n cells are connected in series in each row and m identical such rows are connected in parallel:

• Total number of cells = m n.
• Emf of each row = n E, and since the rows are in parallel the net emf of the battery = n E.
• Internal resistance of each row = n r. m rows in parallel give total internal resistance r_eq = n r / m.
• Total resistance of the circuit = (n r / m) + R.

If the current in the external resistance is I, then

Solving for I gives:

From the derived expression it follows that I is maximum when the derivative condition leads to

The bracketed square term [√(n r) - √(m R)]² is minimum (and equals zero) when

m R = n r or R=nr/m

Note: In a mixed grouping of cells, the current through the external resistance is maximum when the total internal resistance of the battery equals the external resistance (r_eq = R).

Since power delivered to the load is P = I² R, the power delivered to the load is also maximum when R=nr/m

Worked examples - circuits with combinations

Example 4. Find the current in the loop.

Sol.

The given circuit can be simplified by combining series and parallel resistances as shown:

After simplification, the resultant emf and resistance give 7/3 A

Therefore the current is 7/3 A

Example 5. Find the emf and internal resistance of a single battery equivalent to the combination of three batteries shown in the figure.

Sol.

Batteries B and C are connected in parallel but with opposite polarities; compute their equivalent emf and internal resistance accordingly.

r_BC = 1 Ω

Now, 

r_ABC = 2 Ω

Summary (optional)

• Emf is work done per unit charge by the non-electrostatic force inside a cell; open-circuit emf equals terminal potential difference.
• Internal resistance reduces the terminal voltage under load and limits current delivered by a cell or battery.
• Series connection increases net emf and adds internal resistances; parallel connection keeps emf same but reduces internal resistance.
• Mixed combinations allow tailoring of emf and internal resistance; maximum current or power delivery to a given load occurs when the total internal resistance equals the external resistance.