RC Circuits RC Circuits Charging At t=0, capacitance is uncharged - PDF document

RC Circuits

RC Circuits – Charging At t=0, capacitance is uncharged and C Q=0 (initial condition). At t=0, switched is closed, it the capacitor has no charge, it behaves like a conductor and I=  /R.  R After the capacitor is completely charged, Q=C  ,  V C =  and  V R =0. I=0 and the capacitors behave like an insulator.

RC Circuits – Charging q q d q        IR R C C d t C    CR dq (C - q) dt dq 1   - dt Integration constant  q - C CR t      n( q - C ) - K'  CR R t -     K' q - C Ke CR (K e ) t -     q C K e CR          At t 0, q 0 0 C K K - C t -     q C ( 1 e CR )   t t dq C - -    I e CR e CR d t CR R t -     V IR e CR R  V R +  V C =  t q -     V (1 - e CR ) C C

RC time constant  =RC is known as the RC time constant. It indicates the response time (how fast you can charge up the capacitor) of the RC circuit.  t -  I e CR R   I  e 2.72 R -  1 e 0 . 37   - 1 I e R  ~ 0 . 37 R t t=RC t -    q C ( 1 e CR ) q   C  2 1.414    -1 q ( 1 e ) C 1  ~ 0 . 63 C  0 . 707 2 t Nothing to do with RC circuits t=RC

RC Circuits – Discharging At t=0, capacitance is charged with a charge Q (initial condition). At t=0, switched is closed, the capacitor starts to discharge. C R After the capacitor is completely discharged, Q=0,  V C = 0,  V R =0 and I=0.

RC Circuits – Discharging q q d q d q       0 IR R 0 (I - ) C C d t d t   CR dq - q dt dq 1   Integration constant - dt q CR t    C  n q - K' R CR t -    K' q Ke CR (K e ) t -   q K e CR     At t 0, q Q Q K t -   q Qe CR t dq Q -    I e CR d t RC t Q -     V IR e CR R C  V R +  V C = 0 t q Q -    V e CR C C C

In Summary For both charge and discharge, Q, I,  V C , and  V R must be one of the following two cases: y y 0 t - y  y 0 e RC t y y  t -  y y (1 - e RC )  y can be Q, I,  V C , or  V R t

Class 26: Magnetic force acting on a moving point charge

Magnetic Field 1. All single magnets have two poles, N and S. 2. Externally, magnetic field lines come out from the N pole and getting into the S pole. 3. Between two magnets, like poles repel and unlike poles attract. 4. The geographical north pole of earth is actually the S pole of a bar magnet. 5. We will explain why there is magnetic field later.

From class 3 cross product between two vectors    A B Direction:  B   Magnitude: A        A B | A || B | sin ˆ ˆ ˆ i j k   A A A A A A ˆ ˆ ˆ      y z x y x z A B A A A i j k x y z B B B B B B y z x y x z B B B x y z a b  ad - bc c d

From class 3 A common symbol A vector perpendicular and or pointing into the screen /paper. A vector perpendicular and or pointing out of the screen /paper.

Magnetic Force Acting on a Moving Charge When a charge particle moves in a magnetic field B, there will be magnetic force acting on the particle:   B    F B q v B F ` v 1. Unit of magnetic field is Tesla (T). 2. If there is magnetic field, only under two conditions the magnetic force on the charge particle will be zero: (i) the particle is not moving (v=0), or (ii) it is moving in parallel or antiparallel to the magnetic field (sin  =0). 3. The magnetic force is always perpendicular to the magnetic field and the velocity.     F B v 0 4. The magnetic force does no work because . 5. If you want to determine the direction acting on a negative charge particle, treat it like a positive charge first, then reverse the force direction at the end.