Electric Current

Objectives: State the definition of electric current, 𝐼= ∆𝑄 ∆𝑡 ; State the definition of electric resistance, 𝑅= 𝑉 𝐼 ; Appreciate that metallic conductors at a constant temperature satisfy Ohm’s law, 𝐼∝𝑉; Appreciate that the potential drops as one moves across a resistor in the direction of the current; Understand that a resistor dissipates power, P=VI.

Electric Current A moving charge creates an electric current. Electric current is the amount of charge that moves through the cross-sectional area of a wire per unit interval of time: 𝐼= ∆𝑄 ∆𝑡 The unit of current is the ampere (am-peer), and 1 A = 1 𝐶 𝑠 .

Example: Light falling on a metallic surface causes the emission of electrons from the surface at a rate of 2.2 x 1015 per second. What is the current leaving the surface? 𝐼= ∆𝑄 ∆𝑡 = 2.2× 10 15 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 ×−1.602× 10 −19 𝐶 1 𝑠𝑒𝑐𝑜𝑛𝑑 =3.5× 10 −4 𝐴

What is current? Not Current - In a conductor the ‘free’ electrons move randomly, much like gas molecules in a container. They do so with high speeds on the order of 105 m/s. This random motion, however, does NOT result in electric current – as many electrons move to the right as to the left. Current - The presence of an electric field inside the conductor forces the electrons to accelerate in a direction opposite to the electric field (because electrons are negatively charged) and this orderly motion of the electrons in the same direction is what makes electric current.

Conventional current

Why do wires heat up? The electrons increase their kinetic energy as they move through the metal but soon they will suffer inelastic collisions with the atoms of the material, which means they will lose energy to the atoms. The electric field will accelerate the electrons again until the next collision. This process repeats, causing the atoms of the material to vibrate with their gained energy (from the electron collisions), which shows up macroscopically as the increased temperature of the material.

Thermionic Emission An electric current is also produced when a wire is heated so that it begins to emit electrons in a phenomenon known as thermionic emission. If the wire emits N electrons per second, the current leaving the hot wire is I = eN. e is the charge of an electron: 1.602x10-19 C

Law of conservation of charge As a consequence of the law of conservation of charge, it follows that two devices connected in series will take the same current. When a wire comes to a junction, the current splits (not necessarily equally), so that the total current entering it equals the total current leaving it. Also known as the junction rule.

Electric Resistance The electric resistance of a conductor (for example, a wire of a given length) is defined as the potential difference across its ends divided by the current flowing through it: 𝑅= 𝑉 𝐼 The unit of electric resistance is the volt per ampere ( 𝑉 𝐴 ) and this is defined to be the ohm, symbol Ω. The equation above is the definition of resistance.

History In 1826, Georg Ohm discovered that, when the temperature of a metallic conductor was kept constant, the current through the conductor is proportional to the potential difference across it 𝐼∝𝑉. This statement is known as Ohm’s Law. Materials obeying Ohm’s law thus have a constant resistance at constant temperature. A graph of I vs. V gives a straight line through the origin if the material obeys Ohm’s law.

Ohmic vs. non-ohmic devices Most materials obey Ohm’s law at low temperatures, but as temperature increases, deviations from this law are seen. For example, an ordinary light bulb will obey Ohm’s law as long as the current through it is small. As the current is increased, the temperature of the bulb increases and so does the resistance. Other devices, such as the diode, also deviate from Ohm’s law.

Ohmic vs. non-ohmic devices

Factors affecting the resistance of a wire The electric resistance of a wire (at fixed temperature) is proportional to its length L and inversely proportional to the cross-sectional area A: 𝑅∝ 𝐿 𝐴

Example A wire is subjected to a tension so that its length increases by 10% while the volume of the wire stays the same. How does the resistance of the wire change?

Potential drop The defining equation for resistance (𝑅= 𝑉 𝐼 ) can be looked at in the following way. Solving for the potential difference we find V=IR, which says that if a current flows through a resistor, then there MUST be a potential difference across the ends of that resistor given by the formula above. A resistor is thus said to drop the potential.

Example Two resistors are joined as shown. The top resistor receives a current to 3A. What is the current in the other resistor? What is the current that enters at junction A? 3 A 10 Ω B A 30 Ω

Electric Power We saw earlier that whenever an electric charge ∆Q moves from a point A to a point B such that there exists a potential difference between them, work is being done. This work is W = (∆Q)V. Consider a conductor with a potential difference of V across its ends. In moving a charge ∆Q across the conductor in a time ∆t, the power dissipated in the conductor is 𝑃= 𝑤𝑜𝑟𝑘 𝑡𝑖𝑚𝑒 = 𝑉∆𝑄 ∆𝑡 =𝑉𝐼. This power manifests itself as thermal energy and/or work performed by an electrical device. In devices obeying Ohm’s law, we can use 𝑅= 𝑉 𝐼 to rewrite the formula for power in equivalent ways. 𝑃=𝑅 𝐼 2 = 𝑉 2 𝑅 .

Example: A resistor of resistance 12 Ω has a current of 2.0 A flowing through it. How much energy is generated in the resistor in one minute?

Device Ratings Electrical devices are usually rated according to the power that they use. A light bulb rated as 60 W at 220 V means that it will dissipate 60 W when a potential difference of 220 V is applied across its ends. If the potential difference across its ends is anything other than 220 V, the power dissipated will be different from 60 W.

A light bulb rated as 60 W at 220 V has a potential difference of 110 V across its ends. Find the power dissipated in this light bulb.

The cost of electricity Electricity companies charge for electricity according to the amount of energy used by the consumer. A device that is rated at a power value of 60 W, for example an ordinary light bulb, uses 60 J of electricity every second (when connected to the appropriate source of voltage). The energy used by the light bulb over a time of t seconds is thus E = 60t J. In general, for a device of power rating P the energy used in t seconds is 𝐸=𝑃𝑡=𝑉𝐼𝑡.

The cost of electricity Electricity companies find it more convenient to use a different energy unit by which to charge consumers. They use the kilowatt-hour (kW h) as their energy unit, which is defined as the energy used by a device of power rating 1 kW in 1 hour. This means that 1 kWh = 1000 W X 60 m x 60 s = 3.6 x 106 J. If the cost of 1 kW h is, say $0.1, then the cost of operating one 60 W light bulb over a 24 hr period can be calculated as follows: 𝑒𝑛𝑒𝑟𝑔𝑦 𝑢𝑠𝑒𝑑=60 𝑊×24 ℎ=1440 𝑊ℎ≈1.4 𝑘𝑊 ℎ. Hence 𝑐𝑜𝑠𝑡=1.4 𝑘𝑊 ℎ×$0.1=$0.14

A VERY important graphic