Most simply, plates that allow charges to distribute over their area.
Capacitance, C=ϵ0Ad
Loop Rule
ΣV=0
Around any closed loop
Series Circuits
VBatt−V3−V2−V1=0
We know V = IR, so
VBatt−I3R3−I2R2−I1R1=0
Series Circuits
VBatt−I3R3−I2R2−I1R1=0
I is the same everywhere in a series circuit
VBatt−I(R3+R2+R1)=0
Series Circuits
VBatt−I(R3+R2+R1)=0
In series circuits, then…
VBatt−IRTotal=0
RTotal=R1+R2+R3+…
Parallel Circuits
Choose a path 1→8→7→2→1, then,
VBatt−I1R1=0
Parallel Circuits
Choose a different path 1→8→6→3→1
VBatt−I2R2=0
Parallel Circuits
Since VBatt−I2R2=0 and VBatt−I2R2=0
In each branch of a parallel circuit, ΔV must be the same.
V1=V2=V3=VBatt
Parallel Circuits
In a parallel circuit, the current splits at each junction. So the current leaving the battery is ITotal=I1+I2+I3
And, V=IR or I=V/R which can be substituted in.
ITotal=V1/R1+V2/R2+V3/R3
Parallel Circuits
ITotal=V1/R1+V2/R2+V3/R3
But, V is same across all parallel branches, so
V/RTotal=V/R1+V/R2+V/R3V/RTotal=V(1/R1+1/R2+1/R3)
In Parallel branches
1/RTotal=1/R1+1/R2+1/R3+…
Series Circuits
Current is same at all points
ΣΔV=0RTotal=R1+R2+R3+…
Parallel Circuits
Voltage is same across all branches
ITotal=I1+I2+I31/RTotal=1/R1+1/R2+1/R3+…
Capacitors
Capacitors
Charging Capacitor
As capacitor charges, charge builds up on plates of capacitor, creating electric field, building up a potential difference across the capacitor.
Capacitors
Charging Capacitor
Charge on capacitor: Q=VC
Capacitance: C=ϵ(A/d)
Capacitors - Charging
Loop rule still applies, so
VBatt−VCap−VRes=0
Capacitors - Charging
VBatt−VCap−VRes=0
VBatt−Q/C−IR=0
VBatt−Q(t)/C−dQ(t)dtR=0
Capacitors - Charging
VBatt−Q(t)/C−dQ(t)dtR=0
Look at the initial and final conditions.
At t=0,I=I0,Q=Q0,Q0often=0
As t→∞,I→0,Q→VC
Q(t)=Q0(1−e−t/RC)
At time = 0
Q(t)=Q0(1−e−t/RC)dQ(t)dt=Q0RCe−t/RC
at t = 0, Q(0)=Q0, and I0=Q0RC=V/R
As time goes to infinity
Q(t)=Q0(1−e−t/RC)dQ(t)dt=Q0RCe−t/RC
as t→∞,Q(∞)=VC, and I(∞)=I0(1−1)=0
Capacitors - Disharging
Loop rule still applies, so
VCap−VRes=0
Capacitors - Disharging
VCap−VRes=0
Q/C−IR=0
Q(t)/C−dQ(t)dtR=0
Capacitors - Disharging
Q(t)/C−dQ(t)dtR=0
Look at the initial and final conditions.
At t=0,I=I0,Q0=VC
As t→∞,I→0,Q→0
Q(t)=VCe−t/RC
At time = 0
Q(t)=VCe−t/RCdQ(t)dt=VC−RCe−t/RC
at t = 0, Q(0)=VC, and I0=−VCRC=−VC/R
As time goes to infinity
Q(t)=VCe−t/RCdQ(t)dt=VC−RCe−t/RC
as t→∞,Q(∞)=0, and I(∞)=VC/R=0
Resistors and Capacitors
Resistors
Conductors that have different resistivity
Resistivity, ρ=|→E|J=RAlR=ρlA
Capcitors
Most simply, plates that allow charges to distribute over their area.
Capacitance, C=ϵ0Ad
