Electric Currents and Resistance

• The Electric Battery

• Electric Current

• Ohm’s Law: Resistance and Resistors

• Resistivity

• Electric Power

•Microscopic View of Electric Current: Current Density and Drift Velocity

Volta discovered that electricity could be created if dissimilar metals were connected by a conductive solution called an electrolyte.

This is a simple electric cell.

A battery transforms chemical energy into electrical energy.

Chemical reactions within the cell create a potential difference between the terminals by slowly dissolving them. This potential difference can be maintained even if a current is kept flowing, until one or the other terminal is completely dissolved.

The Electric Battery

Several cells connected together make a battery, although now we refer to a single cell as a battery as well.

The Electric Battery

Electric current is the rate of flow of charge through a conductor:

Unit of electric current: the ampere, A:

1 A = 1 C/s.

Electric Current

The instantaneous current is given by:

A complete circuit is one where current can flow all the way around. Note that the schematic drawing doesn’t look much like the physical circuit!

Electric Current

Electric CurrentCurrent is flow of charge.

A steady current of 2.5 A exists in a wire for 4.0 min. (a) How much total charge passed by a given point in the circuit during those 4.0 min? (b) How many electrons would this be?

Electrons in a conductor have large, random speeds just due to their temperature. When a potential difference is applied, the electrons also acquire an average drift velocity, which is generally considerably smaller than the thermal velocity.

Current Density and Drift Velocity

Current Density and Drift Velocity

We define the current density (current per unit area) – this is a convenient concept for relating the microscopic motions of electrons to the macroscopic current:

If the current is not uniform:

.

Charges move with a drift velocity along the wire.

Current Density and Drift Velocity

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Electric Current

How to connect a battery.

What is wrong with each of the schemes shown for lighting a flashlight bulb with a flashlight battery and a single wire?

By convention, current is defined as flowing from + to -. Electrons actually flow in the opposite direction, but not all currents consist of electrons.

Electric Current

Experimentally, it is found that the current in a wire is proportional to the potential difference between its ends:

Ohm’s Law

The ratio of voltage to current is called the resistance:

Ohm’s Law: Resistance and Resistors

In many conductors, the resistance is independent of the voltage; this relationship is called Ohm’s law. Materials that do not follow Ohm’s law are called nonohmic.

Unit of resistance: the ohm, Ω:

1 Ω = 1 V/A.

Ohm’s Law

Ohm’s Law

Current and potential.

Current I enters a resistor R as shown. (a) Is the potential higher at point A or at point B? (b) Is the current greater at point A or at point B?

Ohm’s LawFlashlight bulb resistance.

A small flashlight bulb draws 300 mA from its 1.5-V battery. (a) What is the resistance of the bulb? (b) If the battery becomes weak and the voltage drops to 1.2 V, how would the current change?

Standard resistors are manufactured for use in electric circuits; they are color-coded to indicate their value and precision.

Ohm’s Law

Ohm’s LawThis is the standard resistor color code. Note that the colors from red to violet are in the order they appear in a rainbow.

Some clarifications:

• Batteries maintain a (nearly) constant potential difference; the current varies.

• Resistance is a property of a material or device.

• Current is not a vector but it does have a direction.

• Current and charge do not get used up. Whatever charge goes in one end of a circuit comes out the other end.

Ohm’s Law

The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area:

The constant ρ, the resistivity, is characteristic of the material.

Resistivity

Geometric property

ResistivityThis table gives the resistivity and temperature coefficients of typical conductors, semiconductors, and insulators.

Current Density and Drift Velocity

The electric field inside a current-carrying wire can be found from the relationship between the current, voltage, and resistance. Writing R = ρ l/A, I = jA, and V = El , and substituting in Ohm’s law gives:

Power, as in kinematics, is the energy transformed by a device per unit time:

Electric Power

or

The unit of power is the watt, W.

For ohmic devices, we can make the substitutions:

Electric Power

Electric PowerHeadlights.

Calculate the resistance of a 40-W automobile headlight designed for 12 V.

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What you pay for on your electric bill is not power, but energy – the power consumption multiplied by the time.

We have been measuring energy in joules, but the electric company measures it in kilowatt-hours, kWh:

1 kWh = (1000 W)(3600 s) = 3.60 x 106 J.

Electric Power

Electric PowerElectric heater.

An electric heater draws a steady 15.0 A on a 120-V line. How much power does it require and how much does it cost per month (30 days) if it operates 3.0 h per day and the electric company charges 9.2 cents per kWh?

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Electric PowerLightning bolt.

Lightning is a spectacular example of electric current in a natural phenomenon. There is much variability to lightning bolts, but a typical event can transfer 109 J of energy across a potential difference of perhaps 5 x 107 V during a time interval of about 0.2 s. Use this information to estimate (a) the total amount of charge transferred between cloud and ground, (b) the current in the lightning bolt, and (c) the average power delivered over the 0.2 s.

• A battery is a source of constant potential difference.

• Electric current is the rate of flow of electric charge.

• Conventional current is in the direction that positive charge would flow.

• Resistance is the ratio of voltage to current:

Summary

• Ohmic materials have constant resistance, independent of voltage.

• Resistance is determined by shape and material:

• ρ is the resistivity.

Summary

• Power in an electric circuit:

• Direct current is constant.

• Relation between drift speed and current:

Summary

DC Circuits

• EMF and Terminal Voltage

• Resistors in Series and in Parallel

• Kirchhoff’s Rules

Electric circuit needs battery or generator to produce current – these are called sources of emf (Electromotive force).

Battery is a nearly constant voltage source, but does have a small internal resistance, which reduces the actual voltage from the ideal emf:

EMF and Terminal Voltage

Terminal Voltage emf

This resistance behaves as though it were in series with the emf.

EMF and Terminal Voltage

EMF and Terminal VoltageBattery with internal resistance.

A 65.0-Ω resistor is connected to

the terminals of a battery whose emf is 12.0 V and whose internal resistance is 0.5 Ω. Calculate (a) the current in the circuit, (b) the terminal voltage of the battery, Vab, and (c) the power dissipated in the resistor R and in the battery’s internal resistance r.

A series connection has a single path from the battery, through each circuit element in turn, then back to the battery.

Resistors in Series

•The current through each resistor is the same

•The voltage depends on the resistance.

•The sum of the voltage drops across the resistors equals the battery voltage:

Resistors in Series

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A parallel connection splits the current; the voltage across each resistor is the same:

Resistors in Parallel

The total current is the sum of the currents across each resistor:

Resistors in Parallel

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The voltage across each resistor is the same:

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This gives the reciprocal of the equivalent resistance:

Resistors in Parallel

ResistorsSeries or parallel?

(a) The lightbulbs in the figure are identical. Which configuration produces more light? (b) Which way do you think the headlights of a car are wired? Ignore change of filament resistance R with current.

ResistorsAn illuminating surprise.

A 100-W, 120-V lightbulb and a 60-W, 120-V lightbulb are connected in two different ways as shown. In each case, which bulb glows more brightly? Ignore change of filament resistance with current (and temperature).

ResistorsCircuit with series and parallel resistors.

How much current is drawn from the battery shown? What is the current through each of the resistor?

Resistors in Series and in Parallel

Bulb brightness in a circuit.

The circuit shown has three identical lightbulbs, each of resistance R. (a) When switch S is closed, how will the brightness of bulbs A and B compare with that of bulb C? (b) What happens when switch S is opened? Use a minimum of mathematics in your answers.

Resistors

Analyzing a circuit.

A 9.0-V battery whose internal resistance r is 0.50 Ω is connected in the circuit shown. (a) How much current is drawn from the battery? (b) What is the terminal voltage of the battery? (c) What is the current in the 6.0-Ω resistor?

a b

c d

Some circuits cannot be broken down into series and parallel connections. For these circuits we use Kirchhoff’s rules.

Kirchhoff’s Rules

Junction rule: The sum of currents entering a junction equals the sum of the currents leaving it.

Kirchhoff’s Rules

Loop rule: The sum of the changes in potential around a closed loop is zero.

Kirchhoff’s Rules

Kirchhoff’s Rules• Junction rule:

• Loop rule:

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1. Label each current, including its direction.

2. Identify unknowns.

3. Apply junction and loop rules; you will need as many independent equations as there are unknowns.

4.Solve the equations, being careful with signs. If the solution for a current is negative, that current is in the opposite direction from the one you have chosen.

Kirchhoff’s Rules

Kirchhoff’s RulesUsing Kirchhoff’s rules.

Calculate the currents I1, I2, and I3 in the three branches of the circuit in the figure.

Calculate the equivalentresistance:

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Solving the coupled equations and express I1 and I2 in terms of I, R1, R2 and R3

Solution:

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• A source of emf transforms energy from some other form to electrical energy.

• A battery is a source of emf in parallel with an internal resistance.

• Resistors in series:

Summary

• Resistors in parallel:

• Kirchhoff’s rules:

1.Sum of currents entering a junction equals sum of currents leaving it.

2.Total potential difference around closed loop is zero.

Summary

• RC circuit has a characteristic time constant:

•Ammeter: measures current.

• Voltmeter: measures voltage.

Summary