Electricity

Internal resistance

* WHEN CURRENT FLOWS it creates a voltage drop across the internal resistance, so less voltage available to external circuit than when current was off.* (voltage not = to EMF) *energy being converted to heat* increasing internal resistance more energy changed to heat ∴ *voltage & current delivered to external circuit decreased* internal resistance increases as battery goes flat.

Phase VL &Vr

*inductor voltage leads the resistor voltage by 90°*

Inductor

*stores energy in the magnetic field.*

Phase Vc & Vr

*the resistor voltage leaves the capacitor voltage by 90°*

Factors affecting the reactance of an inductor

- *directly proportional to the inductance*. bc greater inductance will cause greater opposing voltage ∴ smaller current ∴ larger reactance. - *directly proportional to frequency*. bc higher frequency means current is changing at a greater rate ∴ opposing voltage is greater ∴ smaller current ∴ larger reactance.

Factors affecting the reactance of a capacitor

- *inversely proportional to the capacitance*. bc large capacitance stores more charge ∴ more current in the circuit as the plates are charged & discharged. ∴ low reactance. - *inversely proportional to the frequency*. bc high frequency means more charge is flowing on and off the capacitor plates at a high rate ∴ current is high. ∴ low reactance.

How to increase inductance :

- adding more coils - using core with higher permitivity - larger coil area

Kirchhoff's Current Law

- at any junction in a circuit, the total current entering the junction equals the total current leaving. (*total current = zero A*) - *conservation of electric charge*.

Capacitors connected in series

- charge on each capacitor is the same. - voltage across each capacitor add to give the supply voltage. - combined capacitance is given by 1/CT = (1/C1 + 1/C2...)

Series circuit

- current has the same value at each part of the circuit. - voltages across the series resistors add up to the supply voltage. - Rt= R1 + R2

Parallel circuits

- current in each branch sum to total current. - voltage is the same across all BRANCHES in parallel as supply voltage. - Rt= (1/R1 + 1/R2)-1

Kirchhoff's Voltage Law

- the *total voltage* about any *closed loop* in the circuit is *zero* volts. - conservation of energy.

Capacitors connected in parallel

- the total charge stored is the sum of the charge on each capacitor. - voltage across the capacitor is the same as the voltage of the power source. - combined capacitance is given by Ct= C1 + C2...

AC in a capacitor

AC causes the plates to become alternately charged. so there is a movement of charges back and forth is the circuit ; *making the lamp glow continuously*.

The current during charging

At any instant during the charging of a capacitor, the voltages across the capacitor and the resistor must add to the voltage of the supply. Vs= Vr +Vc Therefore as Vc increases during charge, Vr must decrease. *Because it is the voltage across the resistor that determine the current in the circuit* (according to Ohms Law *I=Vr/R*) ; as Vc increases the current must decrease.

Energy stored in a capacitor and inductor when charging VS when discharging

CHARGING : - when capacitor fully charge energy stored in electric field of capacitor (E=1/2QV) - when capacitor fully charge I=0A ∴ no energy stored in inductor (E=1/2LI^2) DISCHARGING : - when capacitor discharging charge (voltage) is decreasing ∴ energy stored by electric field of capacitor decreases. (E=1/2QV) - when capacitor discharging current increases ∴ energy stored by magnetic field of inductor increases. (E=1/2LI^2) -- *energy stored by magnetic field of inductor comes from the drop in energy stored in electric field of capacitor*

Lenz's law

Lenz's law states that as a consequence of this '*an induced EMF opposes the change in the magnetic field producing it*'.

Step-down transformers

Np > Ns Vs output is very small but current can be high.

Step-up transformers

Ns > Np induces a large Vs. less power is 'wasted' as heat when transferred at high voltages as current is less.

Power

Power : the rate at which energy is transferred. P=IV P= I^2R P=V^2/R

Why Vc = Vs when fully charged?

Q=CV as Q increases V increases (directly prop). bc plates (equally and oppositely) charged, will have potential difference across. *this voltage will increase until it matches supply voltage*. Vr= 0 as *no current flows & KVL states that total voltage about any closed loop = 0 V meaning Vc= Vs

Time constant for an inductor

T = L/R the time taken for the current/voltage to change by 63% of its total change.

Ohms law

V=IR

Inductors: voltage-time graph switch open

VL = IR VL directly prop to current so same *exponential decay*

Magnetic flux

a measure of the magnetic field strength over a given area. (Φ= B x A) in Wb MIN: when magnetic flux flows parallel to surface. MAX: when magnetic flux flows at right angles.

Inductors: voltage-time graph switch closed

an exponential decay curve. *Vs = VL+ IR* due to KVL at the start the current is changing at its greatest rate, so the inductor voltage is maximum. when the current reaches its max value it stops changing and so the inductor voltage is zero. *current is only limited by resistance so I at max*. at start VL at max so Vr=0 VL=Vs as current increases VL decreases while Vr increases [bulb brightness]

Magnetic field B

are produced whenever a current flows. bigger current= bigger magnetic field. magnetic field strength in Tesla, T

Inductors: switch closed - why bulb goes dimmer?

at start rate of change of current (and thus magnetic flux) is large = creates large back EMF that opposes the current ∴ *most of current goes through bulb* ∴ increased *power output = brighter bulb*. as rate of change of current decreases =smaller back EMF induced ∴ *its effective resistance decreases* ∴ *more current through inductor and less through bulb ∴ reduced power output= dimmer*

Factors determining capacitance

capacitance/*ability to store charge* is - directly proportional to *area* of plates. - inversely proportional to the *distance* separating plates. - directly proportional to the*dielectric constant* of the insulating material between the plates.

RC circuits

circuit consisting of a resistor connected in series with a capacitor.

Capacitor

components which store *electric charge* and electric potential energy. made up of two parallel metal plates of overlapping area, separated by a layer of insulating material (dielectric). capacitors can be charged by connecting to the terminals of a battery inducing a voltage across the capacitor.

Transformer

consists of two coils wound onto an iron core. when an AC supply is connected to the primary coil, the changing current in the circuit creates a changing magnetic flux in the iron core. this changing flux passes through the secondary coil, where a voltage is induced. *more coils = larger induced voltage*.

AC circuits

current & voltages regularly change directions.

Electric current direction

current flows from higher potential to lower potential (eg start at positive terminal of 11V vs 10V battery as has higher potential).

Energy stored in an inductor

current in an inductor causes a magnetic field to form. energy is stored in the magnetic field of an inductor. E=1/2LI^2

Self-inductance: when switch is closed

current increases from 0 to a steady max value. current is increasing, there will be changing magnetic flux in the coil and therefore an induced back EMF is produced that opposes the change in magnetic field producing it ; by delaying the build-up of current in the coil*. ∴ VL opposes Vs to prevent current from increasing rapidly. KVL: *Vs = VL + Vr* when the switch is closed, the circuit forms a closed loop and so the induced voltage cannot exceed the applied voltage (Kirchhoff's voltage law). max VL= Vs bc at start I=0 ∴ Vr=0

The current during discharging

during discharging the voltage source is removed so the capacitor is the only source of voltage in the circuit. the voltage across the resistor is therefore always equal to the voltage across the capacitor. *bc it is the voltage across the resistor that determine the current in the circuit* (according to Ohms Law I=Vr/R) as Vc (and thus Vr) decreases current decreases.

Electromagnetic induction

electromagnetic induction involves the conversion of *mechanical energy to electrical energy*. induction involves a *changing magnetic field* in or near a circuit or coil. An induced voltage is produced which can cause a current if the circuit is complete. eg a magnet moving in and out of a solenoid will induce a current.

Energy stored in a capacitor

energy (from voltage source) stored as *EPE by charges in electric field when capacitor fully charged. *half the energy of the battery is stored in capacitor* and other half transferred as heat/ light dueto the resistance of the circuit. Ep = 1/2QV Ep= 1/2CV^2 Ep= 1/2Q^2/C

Voltage

energy change on each coulomb of charge as it moves between two points. V = E/q

V = -L ΔI/Δt

for a voltage to be induced must be a rate of change of current.

Dielectric constant (Er)

gives the *proportion* by which the capacitance increases when the dielectric is placed between the plates, as opposed to if there was air(or a vacuum) between the plates. Er = C with dielectric /C with air or vacuum

Inductors: current-time graph switch closed

growth in current is exponential. due to back EMF opposing build up of current

Reactant LCR

if Xc much bigger than XL Z increases ∴ I decreases

Phase: current & supply voltage

in a resistor : Vr=IR ∴ *the current will be in phase with the resistor voltage* *Vl leads I by 90° I leads Vc by 90°* *Vs²=Vc/L² + Vr²*

Capacitance

is the amount of charge a capacitor can store when connected across a voltage of 1 volt. C= Q/V in Farad F microfarad μF = 1x10^-6 nanofarad nF = 1x10^-9 picofarad pF = 1x10^-12

Impedance

limits current the combined effect of resistance and reactance. *Z= √R² + Xc/L²*

Resistance

limits current flow.

Loop being pushed through a magnetic field

loop is cutting across the magnetic field ∴ charges in loop also moving through the magnetic field and therefore will experience a force. as the loop moves through the magnetic field opposite charges experience a force in opposite directions accumulating on opposite ends. This charge separation requires work and therefore a potential difference (voltage) is induced. because this a closed loop, a current is also induced in the *anticlockwise direction*, as the loop is pushed into a magnetic field.

F = BIL

magnetic force on a *current-carrying* wire in a magnetic field.

F= Bqv

magnetic force on a charge moving in a magnetic field.

Voltage across horizontal wires of loop...

no voltage induced across these wires. because the movement of the wire causes force on the electrons that is *across* the wire rather than along the wire and therefore charges accumulate at *same points* on vertical ends of wires.

How does connecting a resistor in parallel protect against high voltage sparks ?

provides low resistance pathway for the current to travel through. low resistance means a larger time constant T=L/R reduced rate of current change the induced EMF is smaller.

RMS value of AC voltage & current

root mean squared voltage.averae voltage. the equivalent amount of DC voltage & current that delivers same average power output. used bc *average AC voltage or current = 0* (average of a sin function overtime = 0)

Vc= IXc

shows that the capacitor voltage is directly proportional to the current.

Vl=IXl

shows that the inductor volage is directly proportional to the current.

V=BvL

size of induced voltage across a wire moving through a magnetic field.

Transformer : energy

some energy is dissipated as heat. in an ideal transformer: *power (primary)=power (secondary)*. VpIp = VsIs Vs doubles Is halves in order for power in=power out.

Faraday's law

states that '*the size of the induced voltage in a conductor equals the rate of change of magnetic flux.*' V=-ΔΦ/t the (-)tive sign shows Lenz's law. the area of the loop in the magnetic field changes (increases) as the loop moves down, therefore,*flux is changing*(increasing) and therefore a voltage is induced across it. when the loop is completely inside the magnetic field, the area of the loop in the field is no longer changing ∴ the rate of change of flux is 0 and thus no voltage is induced.

Lenz's law: magnet moved towards coil (N pole)

the *induced current* in the coil creates a north pole to repel the magnet and oppose its movement towards the coil.

Self-inductance: when switch is open

the current falls from its max value to 0 open switch creates very large R (theoretically infinite) ∴ time constant decreases so rate of change of current is high there will be a rapid change in magnetic flux in the coil and therefore *a large back EMF is produced that opposes the change in magnetic field (current) causing it; by preventing the current from falling*. (reduce rate of decrease) ∴ the direction of the induced voltage is with KVL : Vl = IR when the switch is open the loop is *no longer closed* and so the induced voltage is no longer fixed to the applied voltage and can reach very high values. induced EMF large enough to produce a *spark*

Resonance

the current reaches a maximum at the resonant frequency. (when current mac = greatest sound, brightest bulb) bc at the resonant frequency *XL=Xc [reactance of inductor & capacitor 180° out of phase so cancel out] ∴ impedance in the circuit = resistance only (Z=R) it is the min Z value so current max. VL & Vc are equal & opposite phase (180° degrees out of phase)∴ cancel each other ∴ Vs=Vr.

Lenz's law: magnet moved away from coil (N pole)

the induced current in the coil creates a south pole to attract the magnet and try to prevent it moving away (oppose its movement away from the coil.)

As the loop leaves the magnetic field...

the induced current is clockwise,opposite to the direction on entering because the *current is due to the voltage induced across the other vertical wire.*

Reactance

the property of a capacitor or inductor to *limit the alternating current* in a circuit. increasing reactance increases impedence so current decreases.

While the loop is completely in the magnetic field...

there is no induced current. The voltage induced across one vertical wire is balanced by an *equal and opposite voltage* induced across the other wire parallel to this. the 2 voltages *'push'* the current in opposite directions and therefore cancel out to give *no induced current*.

Time constant

time, in seconds, for the capacitor voltage or current to *change by 63%* of its total change. T=RC increase R or C increases T s After *5* time constants, the capacitor is considered to be fully charged or discharged and *no current flows*. *Vc = Vbat - (Vbat x 0.37^n)*

Kirchhoff's Laws

used for parallel circuits when more than one of its branches has a voltage supply.

Terminal voltage

voltage when current flows.

EMF

voltage when no current flows.

AC in an inductor

when an inductor connected into an AC circuit; bc current (thus flux) is continually changing there will be a continual voltage induced across the inductor that acts against the continually changing current (Lenz's law) [limit amount of current.

Inductors: current-time graph switch open

when switch is open = open circuit ∴ &current drops to 0 instantaneously*

EMF & internal resistance

when the resistance of the rheostat is decreased, the current increases and so more energy is changed to heat by the internal resistance. This means that the voltage available to the external circuit will decrease. *EMF= V (to external circuit)+ Ir (internal resistance)*

Capacitors : charging

when the switch is closed: Q=CV C constant ∴ Q directly prop to V at start V=0 as capacitor is uncharged. current at max initially so charges flow on to capacitor fast initially ∴ voltages rise rapidly. Vc increases opposing source voltage it ∴ becomes harder to add charge onto a charged plate due to repulsive forces & current has decreased ∴ charges flow onto capacitor more slowly. bc plates (equally and oppositely) charged, will have potential difference across. *this voltage will increase until it matches supply voltage*. when capacitor fully charged current=0 ∴ Vr=0 therefore Vc=Vs (KCL) ∴ voltage across a capacitor shows an *exponential increase* over time.

Capacitors : discharging

when the switch is open: Q=CV Q directly prop to V capacitor voltage drops quickly at first then more slowly as fewer charges remain on the capacitor ∴*exponential decay* over time.