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## Resistance

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**Review of Resistors**• The resistance is an intrinsic property of a material which impedes the flow of charge requiring a pd to be applied so that there can be current flow.**Review of Resistors**• The resistance is an intrinsic property of a material which impedes the flow of charge requiring a pd to be applied so that there can be current flow. • From ohm’s law, the resistance of a device is the ratio of the potential difference across it to the current flowing through it.**RC Circuits**• The current in the previous circuits are time independent once the emf of the source is time independent.**RC Circuits**• The current in the previous circuits are time independent once the emf of the source is time independent. • However we may have circuits which are time dependent. • An example is an RC circuit.**A RC circuit consists of a resistor R connected in series**with a capacitor C.**The following circuit can be use the test the charging and**discharging of the capacitor through the resistor.**Consider charging:**• Initially the capacitor is uncharged.**Consider charging:**• Initially the capacitor is uncharged. • When in the charging position current flows and the capacitor charges. • From Kirchoff’s law:**Which can be written as:**• Since • We can rewrite the equation as,**Which can be written as:**• Since • We can rewrite the equation as, • Doing some algebra,**Which can be written as:**• Since • We can rewrite the equation as, • Doing some algebra, • We must separate the variables so that we can integrate and find the final charge on the capacitor.**Separating variables,**• Integrating,**Separating variables,**• Integrating,**Separating variables,**• Integrating, • Which gives,**q(t)**VbatC t • Taking the antilog and simplifying we get,**The product RC in the previous equation is called the time**constant. • Has units of time. • Time taken for the charge to increase from zero to 63% of its final value.**Vc**Vbat t • The pd across the capacitor • Which gives**The current for the charging**• Which gives I(t) Vbat/R t**Consider discharging:**• For the discharge position, the battery is no longer in the circuit.**Since**• We can write that**Since**• We can write that • Separating variables,**Since**• We can write that • Separating variables, • Which in separated form is,**Integrating,**• We get • Which after simplification is,**This can be written as, , noting that**the initial charge is CVbat.**This can be written as, , noting that**the initial charge is CVbat. • Differentiating gives the current, • The voltage across the capacitor is,**Limiting conditions:**• At t=0, q= CVbat. • At t=inf, q= 0. q CVbat t**t**I(t) Vbat t**Power**• The net rate of energy transfer from the source (battery) P is given by, • Power is in watts(W) or joules/second • The rate at which energy is dissipated through through the resistor is, • The energy lost is in the form of thermal energy. • The power supplied to the capacitor is,**Energy**• The total energy supplied by the battery in a time t is given by, • The total energy dissipated in a time t, • The total energy supplied to the capacitor in time t,**Energy**• From the conservation of energy,**From the conservation of energy,**• where,**From the conservation of energy,**• where,**From the conservation of energy,**• where,