Internal Battery Resistance
If the internal resistance is zero, the terminal voltage equals the emf
In a real battery, there is internal resistance, r
The terminal voltage, DVab = e – Ir
Definition: For an Ideal Battery, the internal resistance = 0Ω
Resistors in Series
ΔV = I R1 + I R2 = I (R1+R2)
Req = R1 + R2 +…
Resistors in Parallel
In parallel, Req is always less than the smallest resistor in the group.
Kirchhoff’s Rules
Junction Rule:
The sum of the currents at any junction must equal zero
Currents directed into the junction are entered into the -equation as +I and those leaving as -I
A statement of Conservation of Charge
Loop Rule:
The sum of the potential differences across all elements around any closed circuit loop must be zero
A statement of Conservation of Energy
Kirchhoff’s steps in solving a problem
Step 1: choose and mark the loops from the labelled junctions.
Step 2: choose and mark current directions. Mark the potential change on resistors.
Step 3: apply Junction rule
Step 4: apply Loop rule
Step 5: solve the equations for unknowns.
Capacitor and Capacitance
Capacitors are devices that store electric charge;
A capacitor consists of two conductors separated by an insulator:
These conductors are called plates;
A charged capacitor has an equal amount of charge on each plate, but of opposite signs.
A potential difference develops between the plates when the capacitor is charged.
Parallel plate capacitor
Each plate is connected to a terminal of a battery;
The battery creates an electric field, electrons move, plates are charged;
After the capacitor is charged, the potential difference across the capacitor plates is the same as that between the terminals of the battery.
Electric field and potential in the capacitor
The electric field between two large parallel plates of area A is given by :
where σ is surface charge density
The electric potential is related to the electric field by:
The electric field is uniform between two large and charged plates, and directed towards decreasing electric potential.
Capacitance
The capacitance, C, of a capacitor is defined as the ratio of the magnitude of the charge Q on either conductor to the potential difference ΔV between the conductors
The SI unit of capacitance is the farad (1F=1C/V)
If the parallel metallic plates have area A and separation d the capacitance is given by:
Capacitance is always a positive quantity;
The capacitance of a given capacitor is constant;
The capacitance is a measure of how much charge can be stored by the capacitor per unit of potential difference;
The farad is a large unit. Capacitors typically have capacitances measured in microfarads (mF) and picofarads (pF)
Capacitors in parallel
W
hen
capacitors are first connected in the circuit, electrons are
transferred from the left plates through the battery to the right
plates;The left plates are positively charged and the right plates are negatively charged when the capacitors are charged.
The potential difference across the capacitors is the same and it is equal to the voltage of the battery: ΔV1 = ΔV2 = ΔV
The capacitors reach their maximum charge when the flow of charge stops;
The total charge collected on the capacitors’ plates is equal to the sum of the charges on the capacitors:
Qtotal = Q1 + Q2
The capacitors can be replaced with one capacitor with a capacitance:
Ceq = C1 + C2
The equivalent capacitance of a parallel combination of capacitors is greater than that of any of the individual capacitors
Capacitors in Series
When a battery is connected to the circuit, electrons are transferred from the left plate of C1 to the right plate of C2 through the battery
As this negative charge accumulates on the right plate of C2, an equivalent amount of negative charge is forced off the left plate of C2, leaving it with an excess positive charge;
Both right plates end up with a charge –Q, whereas each left plate gains +Q.
The potential differences add up to the battery voltage:
ΔVtot = ΔV1 + ΔV2 + …
The equivalent capacitance Ceq of a series combination is given by:
T
he
equivalent capacitance of a series combination is always less than
that of any individual capacitor in the combination.
Energy Stored in a Charged Capacitor
Assume the capacitor is being charged and, at some point, has a charge q on it;
The work dW needed to transfer a charge dq from one plate to the other is:
So, the total work done is:
The work done in charging the capacitor is converted to electrical potential energy U :
This applies to a capacitor of any geometry;
The energy stored increases with charge and potential difference;
In practice, there is a limit to the maximum energy that can be stored in a capacitor.
Capacitors in DC circuits
Before charge: Q0 = 0 and i = E/R
Charging a Capacitor (Switch is Open)
Derive q(t), i(t), VC(t) and VR(t)
τ = RC is the circuit time constant
Discharging a Capacitor (Switch Open)
Before discharge: Q0 = CV0 and i = 0
Derive q(t), i(t), VC(t) and VR(t)
Sources of Magnetism
Moving charges
Permanent magnets
Atomic Dipoles
NOTE: There is no experimental evidence for magnetic monopoles
Direction of the magnetic field B is the direction of a tangent to a field line.
A charge q moving with velocity v at an angle θ to a magnetic field B experiences a force:
The magnitude of the magnetic force on q is:
F = q v B sin θ
θ is the angle between v and B
F is always perpendicular to the plane of v and B
F is zero when v and B are parallel (θ = 0o) or anti-parallel (θ =180o)
F is maximum when v and B are perpendicular (θ = 90o)
Magnetic Flux
Magnetic Flux Φ, (unit Weber -Wb) is the total number of magnetic lines passing through an area A at an angle θ to the normal of area A.
For a uniform B field and a constant Area A:
General form of equation
Note: that the flux is the scalar product of B and A
The flux density gives the magnitude of the B field. This expresses the number of lines passing through a unit area normal to the lines, i.e. when θ = 00
The SI unit of magnetic field is the Tesla
(T):
A non - SI , commonly used unit is the Gauss (G): 1 T = 104 G
Force on a moving charge - Circular path of a charge moving at right angles to a uniform B field
Therefore:
Force on a current carrying straight wire
Suppose +q passes through a wire. If its speed is v, then in time t it travels L = v · t. But i = q/t. Thus q ∙ v = i · L , and F = q v × B becomes:
Magnitude of F = i L B sin φ
Magnetic field magnitude due to current carrying wire
Force between two current-carrying wires ( Derivation )
Wire 2 is in field B1 due to wire 1.
Therefore the force F12 that wire 2 experiences per unit length is:
F12 will be repulsive for currents in opposite directions.
Charged particle in E and B fields
A
charged particles with velocity v,
moving through regions with either magnetic or electric field:
