Course Updates - - PowerPoint PPT Presentation

Course Updates

http://www.phys.hawaii.edu/~varner/PHYS272-Spr10/physics272.html

Notes for today: 1) Complete Chap 26, Problem session Friday 2)Assignment 6 (Mastering Physics) online and separate, written problems due Monday 3)Quiz 3 on Friday 4)Review Midterm 1

Kirchhoff’s Rules

Kirchhoff’s rules are statements used to solve for currents and voltages in complicated circuits. The rules are Rule I. Sum of currents into any junction is zero.

∑ =

i i

I

Rule II. Sum of potential differences in any loop is zero. (This includes emfs)

∑ =

i i

V

a b d c

= + + +

da cd bc ab

V V V V

12 2 1

I I I = +

Why? Since charge is conserved. Why? Since potential (energy) is conserved

a b

ε R C I I

( )

q C e t RC = −

−

ε 1

/

t q

RC 2RC Cε

C

a b

+

• -

ε R + I I

RC Circuits

q

RC 2RC

t

q C e t RC =

−

ε

/

Cε

RC Circuits

(Time-varying currents, charging)

• Charge capacitor:

C initially uncharged; connect switch to a at t=0

• Loop theorem ⇒
• Convert to differential equation for Q:

a b

ε R C I I

Calculate current and charge as function of time.

dt dQ I = C Q dt dQ R + = ε

⇒ Would it matter where R is placed in the loop??

Q IR C ε − − =

• Guess solution:
• Check that it is a solution:

= ⇒ = Q t

ε C Q t = ⇒ ∞ =

Note that this “guess” fits the boundary conditions:

a b

ε R C I I

dQ Q R dt C ε = +

• Charge capacitor:

/

1

t RC

dQ C e dt RC ε

−

⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠

ε ε ε = − + − = +

− −

) 1 (

/ RC t RC t

e e C Q dt dQ R

⇒

!

(1 )

t RC

Q C e ε

−

= −

Charging Capacitor

Charging Capacitor

• Current is found from

differentiation:

• Charge capacitor:

a b

ε R C I I

( )

/

1

t RC

Q C e ε

−

= −

/ t RC

dQ I e dt R ε

−

= =

⇒

Conclusion:

• Capacitor reaches its final

charge(Q=Cε ) exponentially with time constant τ = RC.

• Current decays from max

(=ε /R) with same time constant.

Charging Capacitor

Q

Cε

Charge on C Max = Cε 63% Max at t = RC

( )

/

1

t RC

Q C e ε

−

= −

t

RC 2RC

I t

ε /R

/ t RC

dQ I e dt R ε

−

= =

Current Max = ε /R 37% Max at t = RC

Discharging Capacitor

• Guess solution:
• Check that it is a solution:

/

1 e

t RC

dQ C dt RC ε

−

⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠

ε C Q t = ⇒ = 0 = ⇒ ∞ = Q t

Note that this “guess” fits the boundary conditions:

C

a b

+ +

• -

ε R

dQ Q R dt C + =

I I ⇒ !

R dQ dt Q C e e

t RC t RC

+ = − + =

− −

ε ε

/ /

/ / 0e

e

t t RC

Q Q C

τ

ε

− −

= =

Conclusion:

• Capacitor discharges

exponentially with time constant τ = RC

• Current decays from initial max

value (= -ε/R) with same time constant

• Discharge capacitor:

a

C

b

+

• -

ε R + I I

⇒

• Current is found from

differentiation:

/ t RC

dQ I e dt R ε

−

= = −

Minus sign: Current is opposite to

• riginal definition,

i.e., charges flow away from capacitor.

/ / 0e

e

t t RC

Q Q C

τ

ε

− −

= =

Discharging Capacitor

Discharging Capacitor

Charge on C Max = Cε 37% Max at t = RC

/ t RC

dQ I e dt R ε

−

= = −

Current “Max” = -ε/R 37% Max at t = RC

t Q

Cε RC 2RC

• ε /R

I t

zero

/

e

t RC

Q Cε

−

=

Midterm 1 statistics

A B C D F

Cumulative statistics

Midterm Review

• Not many systematic, consistent problems
• However, some recurrent themes:

– Electric Potential (V) vs. Potential Energy (U) – Electric field sums as vector quantity – Potential field (V) sums as a scaler – Area of a circle is πr2

• Detailed points breakdown (by roster ID) linked

from course webpage

• Will go over in outline form problems next…

For next time

• HW #6 Assigned due next Monday
• Quiz #3 on Friday
• Problem session Friday prior to quiz