Physics 241 Electricity & Optics Lecture 12 Chapter 25 sec. 6, 26 sec. 1 Fall 217 Semester Professor Koltick
Circuits With Capacitors C Q = C V V = Q C + V R C, Q Kirchhoff s Loop Rule: V I R V = V I R Q C = But if I then Q will increase or decrease. Q is a function of time.
Charge on a Capacitor The net charge on a capacitor is the sum of all charges that have been collected on the plates: Q t = Q + I t dt Fundamental Theorem of Calculus: dq dt = I(t) In circuits with capacitors, we need to specify the initial conditions. t
RC Circuits When t <, the switch is open. The capacitor has an initial charge Q as indicated. How will Q change with time after the switch is closed at t =? R C +Q Q
Lecture 12-5 Discharging a Capacitor in RC Circuits 1. Switch closed at t=. Initially C is fully charged with Q 2. Loop Rule: Q C IR 3. Convert to a differential equation dq Q dq I R dt C dt 4. Solve it! I Q Q e t/ RC dq Q I dt RC e t/ RC
Time Constant The quantity τ = RC is called the time constant. Units: R C = V A C V = C C/s = s It is the characteristic time needed to charge or discharge the circuit. Larger τ means it charges or discharges more slowly.
Lecture 12-6 Charge Q(t) during Discharging Qt () Qe Qt () Qe trc / t/ RC Qt ( ) Qe Qe e Qt ( ) e.37 Qt ( ) ( t)/ t/ 1 1 t Q( ) Qe.37Q C C V A s 1 time constant.37q t Qt () Q (1 )
Lecture 12-7 t / Qt () Qe Current I(t) during Discharging Q Q 1 Q V I I RC R dq dt It Ie e It e It t/ 1 1 ( ) ( ).37 ( ) I I Q t/ e t/ Ie t It () I(1 )
RC Circuits Kirchhoff s Loop Rule: I(t) R Q t C = Differentiate R di dt + 1 I t C = di dt = I(t) RC This is a differential equation R I(t) C +Q Q This reminds you that I is not constant it is a function of t.
RC Circuits How to solve di = I(t)? dt RC Solve by integration: 1 di I(t) dt = 1 RC t di I = 1 RC log t dt I(t) I R = t RC I(t) C +Q Q
RC Circuits log I(t) I = t RC Exponentiate both sides: I t = I e t/rc As with all first order differential equations, we have one constant of integration (I ) that must be determined from the initial conditions. R I(t) C +Q Q
RC Circuits I R Q C = I = Q RC Complete solution: R C +Q I(t) I t = Q RC e t/rc The negative sign means the current flows opposite the direction we assumed for the arrow. Q
RC Circuits Reverse the arrow: I t = Q RC e t/rc This makes sense because current will flow in the direction of decreasing electric potential. I I =.368 I I(t) t = RC R t I(t) C +Q Q The capacitor initially acts like a voltage source but eventually acts like an open circuit.
Charging a Capacitor + V R C V Suppose the capacitor is initially uncharged when the switch is open. The switch is closed at time t =. What is the potential difference across the capacitor as a function of time?
Lecture 12-5 Charging a Capacitor in RC Circuits 1. Switch closed at t= C initially uncharged, thus zero voltage across C. I R / 2. Loop Rule: Q IR C 3. Convert to a differential equation dq dq Q I R dt dt C 4. Solve it! 1 dq QC e, I e dt R t / RC t/ RC (τ=rc is the time constant again)
Lecture 12-9 Charge Q(t) during Charging time constant Qt C e Q e t/ t/ () (1 ) f (1 ) Qf C Q Q e Q 1 () f(1 ).63 f
Lecture 12-1 Current I(t) during Charging Qt Q e t/ () f (1 ) Q f t / t / dq I e I e dt I Q Q f f RC R Qf C I R It It e It 1 ( ) ( ).37 ( )
Charging a Capacitor + V R I(t) C + V Kirchhoff s Loop Rule: V I t R Q C = Differentiate: R di dt + I(t) C = Same as before
Charging a Capacitor + V R I(t) C + V General solution: I t = I e t/rc What s different this time? The initial condition... At t =, the capacitor is uncharged so V = and the initial current is I = V/R.
Charging a Capacitor + V R I(t) C + V Solution for the current: I t = V R e t/rc But what is V? Use Kirchhoff s Loop Rule: V I t R V = V t = V I t R = V 1 e t/rc
Charging a Capacitor + V R I(t) C + V V V =.632 V V(t) t = RC V(t) = V 1 e t/rc The capacitor initially acts like a short circuit (zero resistance) but eventually acts like an open circuit (infinite resistance).
Charging a Capacitor DEMO 9/28/217
Lecture 12-4 Behavior of Capacitors Charging Initially, the capacitor behaves like a wire. After a long time, the capacitor behaves like an open switch. Discharging Initially, the capacitor behaves like a battery. After a long time, the capacitor behaves like an open switch
Lecture 12-1 Capacitor in RC Circuits I I ε charging discharging Switch closed at t=. C initially uncharged, thus V = across C and I = ε/r initially. After a long time, C is fully charged and V C = ε across C and I =. Q Q f (1 e t / RC ) Switch closed at t=. C initially charged, thus V = Q /C across C and I = V /R initially. Q Q e After a long time, C is fully discharged and V C = across C and I =. t/ RC time constant
Lecture 12-12 Energy Conservation in Discharging a Capacitor Discharging: Energy lost by C U QV 1 CV 2 2 2 Power dissipated by R R 2 2 2 t/ () P I t RI e R U W R W R 2 2 / R() t P t dt I R e dt 2 2 2 V R C 1 I R 2 R 2 I Q Q V RC R CV 2 2
Lecture 12-1 Energy Conservation in Charging a Capacitor Charging: Work done by battery Energy stored in C Power dissipated by R 2 W Q C U Q f f 1 2 C 2 2 P I() t RI e R The rest? 2 2 2 t / R Independent of R What if R=? What if R=? W R 2 2 t / R() P t dt I R e dt 2 I R 2 2 RC 1 2 C 2 R 2 2
Lecture 12-8 Behavior of Capacitors Charging Initially, the capacitor behaves like a wire. After a long time, the capacitor behaves like an open switch. Discharging Initially, the capacitor behaves like a battery. After a long time, the capacitor behaves like an open switch
Magnetism Bar magnets always two opposite poles Like poles repel, unlike poles attract We can describe the magnetic field the same way we talked about electric fields. N S S N S N N N S S
Lecture 12-16 Demo - Magnetic Field Lines The magnetic field lines from a permanent bar magnet are shown below Two dimensional computer calculation Three dimensional real-life
Lecture 12-13 Magnetic Field and Magnetic Field Lines Bar magnet... two poles: N and S Like poles repel; Unlike poles attract. Magnetic Field lines: (defined from the direction and density of B similar to the electric field lines are from E) From North to South outside electric dipole
Magnetic Monopoles Can you cut the end off a magnet to get just the North or South poles? S N S N S N Most elementary particles behave like little bar magnets: they have a north end and a south end. Although they are point-like we can still talk about which way their magnetic field is pointing In principle, a fundamental particle with a magnetic charge could exist, but we haven t found any evidence that they do.
Lecture 12-14 Magnetic Field B and Magnetic Force Magnetic force acting on a moving charge q depends on q, v. Vary q and v in the presence of a given magnetic field and measure magnetic force F on the charge We find: F F v qv F qvb This defines B. direction by Right Hand Rule. B is a vector field F v, B F qvbsin (q>) B F N N q v Cm/ s Am T ( tesla) 1 T = 1 4 gauss (earth magnetic field at surface is about.5 gauss)
Lecture 12-15 Magnetic Force on a Current Consider a current-carrying wire in the presence of a magnetic field B. There will be a force on each of the charges moving in the wire. What will be the total force df on a length dl of the wire? Suppose current is made up of n charges/volume each carrying charge q and moving with velocity v through a wire of cross-section A. A Force on each charge = qv B Total force = Current = df n A( dl) qv B I n Avq df Idl B For a straight length of wire L carrying a current I, the force on it is: F ILB
Earth s Magnetic Field The orientation of the earth s magnetic dipole is not constant: From 194 to 1984 it moved 75 km (9.4 km/year) The rate of change is also not constant: From 1973 to 1984 it moved 12 km (11.6 km/year)
Earth s Magnetic Field By convention, the N end of a bar magnet is what points at the Earth s North Geographic Pole. Since opposite poles attract (analogous to opposite electric charges), the North Geomagnetic Pole is in fact a magnetic SOUTH pole, by convention. Confusing, but it s just a convention. Just remember that we define N for bar magnets as pointing to geographic North. 2/21/216 27
Earth s Magnetic Field Magnetic North Pole 1999 2/21/216 28
Earth s Magnetic Field Magnetically Quiet Day Magnetically Disturbed Day 2/21/216 29
Earth s Magnetic Field