1. Direct Current And Resistance Electric Current The Battery Resistance And Ohm’s Law Power Internal Resistance Resistors In Combination RC Circuits Written by Dr. John K. Dayton

2. Electric current is the rate of flow of electric charge. By convention, the direction of electric current is always the direction of net flow of positive charge. Current in conductors is the result of mobile electrons collectively drifting in one direction. Ordinarily these mobile electrons move at high speed in random directions. Collisions keep the velocity distribution perfectly symmetric. If an electric field is applied, the force on the electrons causes the velocity in one direction, that of the force, to increase slightly. Electrons then drift slowly in that direction. This drift velocity is like that of the wind in air and produces the electric current. Note that when the current is the result of moving negative charge, as with electrons, the direction of the current is opposite that of the moving negative charge. I is used to represent electric current. Since current is the rate of flow of charge, it has SI units of C/s known as the Amp, A. ELECTRIC CURRENT:

3. EXAMPLE: A current of 3.0 A flows in a wire. How much charge flows past a fixed point in 10 seconds? Click For Answer NOTE:The convention about current indicates this charge is positive, flowing in a certain direction. However, in common wire the actual movement of charge is in the opposite direction and is negative (moving electrons).

4. A cell is an individual unit that produces a potential difference between two metal terminals. It consists of two metal terminals with an electrolytic material between them. As the terminals are oxidized by chemical reaction with the electrolyte, the terminals become negatively charged, one more than the other. Thus a potential difference occurs between the terminals. A battery is any collection of cells connected together. THE VOLTAIC CELL AND BATTERIES: When cells or batteries are connected in series the individual voltages add together. When the cells are connected in parallel the output current is the sum of the currents through each cell. Electromotive force, emf, E, is any electric potential that can be used to sustain a current. The amp-hour rating of a battery is the product of current and time duration of that current that can be sustained by the battery. A 3amp- hour rating will maintain 3 amps for 1 hour or 1 amp for 3 hours, etc.. The amp-hour rating is a non-SI unit of charge, the total amount of charge that flows through the battery. Symbol for a cell

5. EXAMPLE:A 12V battery is rated 3amp·hours. What is the total charge and energy supplied by this battery? Click For Answer

6. RESISTANCE AND OHM’S LAW: The property of a substance called resistivity is a measure of the material’s ability to inhibit the flow of current. Resistivity is usually designated by the symbol . Resistance to flow of current is designated by R and has SI units called Ohms, . If a substance is in the shape of a cylinder with length l and cross- section area A, then its resistance R is proportional to l and inversely proportional to A. The resistivity of the material out of which the cylinder is made is the constant of proportionality. Resistivity has SI units  ∙m.

7. RESISTANCE AND OHM’S LAW: For most substances resistivity increases with increasing temperature. This increase is non-linear but may be approximated by a linear function.  = the resistivity value at temperature T to be calculated.  o = the known resistivity value at temperature T o. T = the temperature in o C or K at which the resistivity will be calculated. T o = the temperature at which r o is known.  = the temperature coefficient of resistivity.

8. EXAMPLE:A resistor is to be made from a 150m long aluminum wire 0.2mm in diameter. What is the resistor’s resistance at 20 o C and at 2000 o C? Click For Answer

9. RESISTANCE AND OHM’S LAW: The symbol for a resistor in a circuit diagram is: As current flows through a resistor it loses energy. A rolling ball loses kinetic energy as it rolls up a hill but its total energy is constant. Current passing through a resistor actually loses energy. The electric potential on one side of a resistor is higher than on other side. In a sense, the current has to climb a potential hill to cross the resistor. The difference in these potentials is called the potential drop across the resistor and is designated V. The current through the resistor is I, and the resistance of the resistor is R. SI units for these are volts, amps and ohms. They are related by Ohm’s law. [Ohm’s Law]

10. POWER SUPPLIED AND POWER LOSS: The emf of a battery is defined as the energy supplied by the battery per unit charge. Electric charge loses energy as it flows through a resistor.

11. POWER SUPPLIED AND POWER LOSS: In this circuit a battery and resistor (load) are connected in series. Current from the battery flows through wires to and through the resistor and then back to the battery. When it leaves the battery the current is at high energy. On its journey through the circuit this energy is lost and the current returns to the battery at low energy. Neither current nor charge is consumed. Chemical energy within the battery is converted to electric potential energy carried by the current leaving the battery. The electric potential energy is lost as the current crosses the resistor where the energy is converted to some other form, mostly thermal (known as joule heating.) As the current moves through the battery, a chemical reaction pumps the current from low U to high U and the journey is repeated. This will continue until the available chemical energy within the battery is exhausted, being ultimately converted to thermal energy.

12. EXAMPLE:A 15  resistor is connected to a 45V power supply. What current flows through the resistor? What power is supplied to the circuit? What is the power loss by the resistor? Click For Answer

13. INTERNAL RESISTANCE: The voltage difference across the terminals of a battery, V T, is the result of the emf of the battery, E, and its internal resistance, r. The Ir term is a voltage drop across the internal resistance of the battery. Thus, some energy produced by the battery is lost internally as thermal energy. The voltage at the terminals will be less than the emf of the battery.

14. EXAMPLE:A battery is made to operate with an emf of 12V and an internal resistance of 0.1 . What is this battery’s terminal voltage when it supplies a current of 3A? Click For Answer

15. RESISTORS COMBINED IN SERIES: When resistors are connected in series the currents through each are the same. The voltage drops across each resistor add to the drop across the group. POWER:

16. RESISTORS COMBINED IN PARALLEL: When resistors are in parallel the same voltage drop is across each of them.

17. RESISTORS COMBINED IN GENERAL: Series: Parallel: currents are equal. voltage drops are equal.

18. EXAMPLE:R 1 =10  and R 2 =15  are connected in series to a 50V battery. What are the currents through, voltage drops across, and power losses at each resistor? Click For Answer

19. EXAMPLE:R 1 =12  and R 2 =15  are connected in parallel to a 18V battery. What are the currents through, voltage drops and power losses at each resistor? Click For Answer

20. RC CIRCUITS: An RC circuit consists of a battery, switch, resistor and capacitor connected in. series When the switch is open no current flows. When the switch is initially closed the initial current is given by Ohm’s law: I o =V/R As charge builds in the capacitor the voltage across the capacitor also builds, but opposes the battery. As the capacitor is charged to Q=CV the current decreases to zero.

21. CHARGING: t IoIo t QoQo t VoVo Time scale in units of .

22. DISCHARGING: t IoIo Time scale in units of . t QoQo t VoVo

23. RC TIME CONSTANT: The product RC has units of time and is referred to as the RC time constant usually designated by the symbol .  is the time it takes the current to (or from) a capacitor to decrease from I o to I o /e. t IoIo.37I o 

24. EXAMPLE:A resistor of 750,000  is connected to a 3.7  F capacitor and 24V battery in series with an open switch. The capacitor is completely uncharged. How long will it take the capacitor to reach 80% of its maximum charge after the switch is closed? Click For Answer