Chapter 27 Magnetism

When the switch is closed, the capacitor will begin to charge. As it does, the voltage across it increases, and the current through the resistor decreases Circuits Containing Resistor and Capacitor (RC Circuits)

To find the voltage as a function of time, we write the equation for the voltage changes around the loop: Since Q = dI/dt, we can integrate to find the charge as a function of time:

26-5 Circuits Containing Resistor and Capacitor (RC Circuits) The voltage across the capacitor is V C = Q/C: The quantity RC that appears in the exponent is called the time constant of the circuit:

26-5 Circuits Containing Resistor and Capacitor (RC Circuits) The current at any time t can be found by differentiating the charge: τ = RC is the time constant of the RC circuit

26-5 Circuits Containing Resistor and Capacitor (RC Circuits) Example 26-11: RC circuit, with emf. The capacitance in the circuit shown is C = 0.30 μF, the total resistance is 20 kΩ, and the battery emf is 12 V. Determine (a) the time constant, (b) the maximum charge the capacitor could acquire, (c) the time it takes for the charge to reach 99% of this value, (d) the current I when the charge Q is half its maximum value, (e) the maximum current, and (f) the charge Q when the current I is 0.20 its maximum value.

If an isolated charged capacitor is connected across a resistor, it discharges: 26-5 Circuits Containing Resistor and Capacitor (RC Circuits)

Once again, the voltage and current as a function of time can be found from the charge: and

26-5 Circuits Containing Resistor and Capacitor (RC Circuits) Example 26-12: Discharging RC circuit. In the RC circuit shown, the battery has fully charged the capacitor, so Q 0 = C E. Then at t = 0 the switch is thrown from position a to b. The battery emf is 20.0 V, and the capacitance C = 1.02 μF. The current I is observed to decrease to 0.50 of its initial value in 40 μs. (a) What is the value of Q, the charge on the capacitor, at t = 0? (b) What is the value of R? (c) What is Q at t = 60 μs?

26-5 Circuits Containing Resistor and Capacitor (RC Circuits) Example 26-14: Resistor in a turn signal. Estimate the order of magnitude of the resistor in a turn-signal circuit.

An ammeter measures current; a voltmeter measures voltage. Both are based on galvanometers, unless they are digital Ammeters and Voltmeters

The current in a circuit passes through the ammeter; the ammeter should have low resistance so as not to affect the current.

26-7 Ammeters and Voltmeters Example 26-15: Ammeter design. Design an ammeter to read 1.0 A at full scale using a galvanometer with a full-scale sensitivity of 50 μA and a resistance r = 30 Ω. Check if the scale is linear.

A voltmeter should not affect the voltage across the circuit element it is measuring; therefore its resistance should be very large Ammeters and Voltmeters

An ohmmeter measures resistance; it requires a battery to provide a current Ammeters and Voltmeters

Summary: How to connect Meters? An ammeter must be in series with the current it is to measure; A voltmeter must be in parallel with the voltage it is to measure Ammeters and Voltmeters

Magnets have two ends – poles – called north and south. Like poles repel; unlike poles attract Magnets and Magnetic Fields

However, if you cut a magnet in half, you don’t get a north pole and a south pole – you get two smaller magnets Magnets and Magnetic Fields

Magnetic fields can be visualized using magnetic field lines, which are always closed loops Magnets and Magnetic Fields

Magnetic Fields similarities with Electric Fields North and South poles Like poles repel Opposite poles attract Field lines outside the material move from N to S Positive and Negative Charges Like Charges repel Opposite Charges attract Field lines move from + to - Electric Magnetic

Magnetic Fields similarities with Electric Fields Electric Field Magnetic Field tangent to the field lines the strongest where the field lines are the closest tangent to the field lines the strongest where the field lines are the closest

The Earth’s magnetic field is similar to that of a bar magnet. Note that the Earth’s “North Pole” is really a south magnetic pole, as the north ends of magnets are attracted to it Magnets and Magnetic Fields

A uniform magnetic field is constant in magnitude and direction. The field between these two wide poles is nearly uniform Magnets and Magnetic Fields

Experiment shows that an electric current produces a magnetic field. The direction of the field is given by a right-hand rule Electric Currents Produce Magnetic Fields

Here we see the field due to a current loop; the direction is again given by a right-hand rule.

A magnet exerts a force on a current- carrying wire. The direction of the force is given by a right- hand rule Force on an Electric Current in a Magnetic Field; Definition of B

The force on the wire depends on the current, the length of the wire, the magnetic field, and its orientation: This equation defines the magnetic field In vector notation: 27-3 Force on an Electric Current in a Magnetic Field; Definition of B

Unit of B: the tesla, T: 1 T = 1 N/A·m. Another unit sometimes used: the gauss (G): 1 G = T Force on an Electric Current in a Magnetic Field; Definition of B Directions of the Magnetic Field:

27-3 Force on an Electric Current in a Magnetic Field; Definition of B Example 27-1: Magnetic Force on a current-carrying wire. A wire carrying a 30-A current has a length l =12 cm between the pole faces of a magnet at an angle θ = 60°, as shown. The magnetic field is approximately uniform at 0.90 T. We ignore the field beyond the pole pieces. What is the magnitude of the force on the wire?

27-3 Force on an Electric Current in a Magnetic Field; Definition of B Example 27-2: Measuring a magnetic field. A rectangular loop of wire hangs vertically as shown. A magnetic field B is directed horizontally, perpendicular to the wire, and points out of the page at all points. The magnetic field is very nearly uniform along the horizontal portion of wire ab (length l = 10.0 cm) which is near the center of the gap of a large magnet producing the field. The top portion of the wire loop is free of the field. The loop hangs from a balance which measures a downward magnetic force (in addition to the gravitational force) of F = 3.48 x N when the wire carries a current I = A. What is the magnitude of the magnetic field B?

The force on a moving charge is related to the force on a current: Once again, the direction is given by a right-hand rule Force on an Electric Charge Moving in a Magnetic Field

Magnetic Force on a point charge Force on a moving charge Direction: Right hand rule Perpendicular to both and Lay hand along palm toward q  Thumb points along q  Thumb points opposite = out of page Arrow coming at you = into page Arrow leaving you

Magnetic Force on a point charge Direction: RIGHT Hand Rule Perpendicular to both v and B Here Into or Out of the page Run fingers along v, curl them towards B, If q is positive, thumb points along F If q is negative, thumb points opposite F Direction: RIGHT Hand Rule Perpendicular to both v and B Here Into or Out of the page Run fingers along v, curl them towards B, If q is positive, thumb points along F If q is negative, thumb points opposite F

Magnetic Force on a point charge If q is + find the direction of F B v B v B v F F F