Resistance
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Resistance. Review of Resistors. The resis tance is a n intrinsic property of a material which impedes the flow of c harge requiring a pd to be applied so that there can be current flow. Review of Resistors.

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Resistance

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Resistance


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.


  • The unit of the resistor is the ohm ( ).


RC Circuits


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:


  • 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:


  • 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,


  • Separating variables,

  • Integrating,


  • Separating variables,

  • Integrating,


  • Separating variables,

  • Integrating,

  • Which gives,


  • Taking the antilog and simplifying we get,


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:


  • 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,


  • 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, Energy


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,


Resistance in Series and Parallel


  • Series:


  • From the conservation of energy,


  • From the conservation of energy,

  • where,


  • From the conservation of energy,

  • where,


  • From the conservation of energy,

  • where,


  • In general,


  • Parallel:


  • From the conservation of charge,


  • From the conservation of charge,

  • where,


  • From the conservation of charge,

  • where,


  • From the conservation of charge,

  • where,


  • From the conservation of charge,

  • where,


  • In general,


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