Recap of Chapter 25-28

Electric Currents and Resistance

Units
Current	1 A = 1 C/s
Potential difference	1V = 1 J/C
Power	1 W = 1J/s
Resistance	1 $\Omega$ = 1 V/A

• An electric battery serves as a source of nearly constant potential difference.
• Electric current, $I$, refers to the rate of flow of electric charge and is measured in amperes (A): 1 A equals a flow of 1 C/s past a given point.
• The direction of conventional current is that of positive charge flow. In a wire, it is actually negatively charged electrons that move, so they flow in a direction opposite to the conventional current.
• Positive conventional current always flows from a high potential to a low potential.

Ohm's law

• The resistance $R$ of a device is defined through the Ohm’s law

 $V=IR$

• The current $I$ coming from a battery of voltage $V$ depends on the resistance $R$ of the circuit connected to it.
• The resistance $R$ of a wire is inversely proportional to its cross-sectional area $A$, and directly proportional to its length $l$ and to a property of the material called its resistivity $\rho$.

  $R={\frac {\rho l}{A}}$

• Resistivity depends on temperature.

Power

• The rate at which energy is transformed in a resistance R from electric to other forms of energy (such as heat and light) is equal to the product of current and voltage.

  $P=IV$ 

• For Ohmic resistors, Power can also be written as

  $P=I^{2}R={\frac {V^{2}}{R}}$

• SI Unit of power is Watts (W)
• The total electric energy transformed in any device is equals the product of the power and the time during which the device is operated. In SI units, energy is given in joules (1 J = 1 W . s), but electric companies use a larger unit, the kilowatt-hour. (1kWh = $3.6\times 10^{-6}$ J).

Direct and Alternating current

• Electric current can be direct current (dc), in which the current is steady in one direction; or it can be alternating current (ac), in which the current reverses direction at a particular frequency $f$

 $I=I_{0}\sin \omega t$

where $\omega =2\pi f$

• The rms values of sinusoidally alternating currents and voltages are

 $I_{\textrm {rms}}={\frac {I_{0}}{\sqrt {2}}}$ and $V_{\textrm {rms}}={\frac {V_{0}}{\sqrt {2}}}$

• The peakvalues of the current and voltage are $I_{0}$ and $V_{0}$
• The power relationship, $P=IV=I^{2}R={\frac {V^{2}}{R}}$, is valid for the average power in alternating currents when the rms values of V and I are used.

Microscopic view of the current

• Current density ${\vec {j}}$ is the current per cross-sectional area.
• From a microscopic point of view, the current density is related to the number of charge carriers per unit volume, $n$, their charge, $q$, and their drift velocity, ${\vec {v}}_{d}$ , by

  ${\vec {j}}=nq{\vec {v}}_{d}$

• conductivity $\sigma$ is the inverse of resistivity

$\sigma ={\frac {1}{\rho }}$

• The electric field within a wire is related to ${\vec {j}}$

  ${\vec {j}}=\sigma {\vec {E}}$

DC Circuits

emf and internal resistance

• A device that transforms another type of energy into electrical energy is called a source of emf ${\mathcal {E}}$. The emf is the potential difference determined by the chemical reactions in the battery and equals the terminal voltage when no current is drawn.
• A battery behaves like a source of emf in series with an internal resistance.
• When a current is drawn, the voltage at the battery’s terminals is less than its emf by an amount equal to the potential decrease $Ir$ across the internal resistance.

Resistances in series and in parallel

• When resistances are connected in series (end to end in a single linear path), the equivalent resistance is the sum of the individual resistances.

 $R_{\textrm {eq}}=R_{1}+R_{2}+\cdots$

• When resistors are connected in parallel, the reciprocal of the equivalent resistance equals the sum of the reciprocals of the individual resistances:

 ${\frac {1}{R_{\textrm {eq}}}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+\cdots$

Kirchoff's Rules

• Kirchhoff’s junction rule is based on conservation of electric charge and states that the sum of all currents entering any junction equals the sum of all currents leaving that junction.
• Kirchhoff’s loop rule, is based on conservation of energy and states that the algebraic sum of the changes in otential around any closed path of the circuit must be zero.

RC Circuits

• A RC circuit contains a resistor R in series with a capacitance C.

Charging the capacitor

• Charging a capacitor with a DC emf: the voltage across the capacitor rises gradually in time characterized by an exponential of the form

  $V(t)={\mathcal {E}}(1-e^{-t/RC})$

• The time it takes for the voltage to reach 63 percent of its maximum value is called the time constant $\tau$

  $\tau =RC$

• The current through the resistor decreases as

  $I(t)=I_{0}e^{-t/RC}$

Discharging the capacitor

• When discharging, the voltage across the capacitor drops to 37 percent of its initial value in $\tau$

  $\tau =RC$

• The charge and potential difference across the capacitor decreases as

  $V(t)=V_{0}e^{-t/RC}$   $Q(t)=Q_{0}e^{-t/RC}$