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

Resistance


Review of resistors

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 resistors1

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.


Resistance

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


Rc circuits

RC Circuits


Rc circuits1

RC Circuits

  • The current in the previous circuits are time independent once the emf of the source is time independent.


Rc circuits2

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.


Resistance

  • A RC circuit consists of a resistor R connected in series with a capacitor C.


Resistance

  • The following circuit can be use the test the charging and discharging of the capacitor through the resistor.


Resistance

  • Consider charging:


Resistance

  • Consider charging:

  • Initially the capacitor is uncharged.


Resistance

  • Consider charging:

  • Initially the capacitor is uncharged.

  • When in the charging position current flows and the capacitor charges.

  • From Kirchoff’s law:


Resistance

  • Which can be written as:


Resistance

  • Which can be written as:

  • Since

  • We can rewrite the equation as,


Resistance

  • Which can be written as:

  • Since

  • We can rewrite the equation as,

  • Doing some algebra,


Resistance

  • 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.


Resistance

  • Separating variables,


Resistance

  • Separating variables,

  • Integrating,


Resistance

  • Separating variables,

  • Integrating,


Resistance

  • Separating variables,

  • Integrating,

  • Which gives,


Resistance

  • Taking the antilog and simplifying we get,


Resistance

q(t)

VbatC

t

  • Taking the antilog and simplifying we get,


Resistance

  • 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.


Resistance

Vc

Vbat

t

  • The pd across the capacitor

  • Which gives


Resistance

  • The current for the charging

  • Which gives

I(t)

Vbat/R

t


Resistance

  • Consider discharging:


Resistance

  • Consider discharging:

  • For the discharge position, the battery is no longer in the circuit.


Resistance

  • Since

  • We can write that


Resistance

  • Since

  • We can write that

  • Separating variables,


Resistance

  • Since

  • We can write that

  • Separating variables,

  • Which in separated form is,


Resistance

  • Integrating,


Resistance

  • Integrating,

  • We get

  • Which after simplification is,


Resistance

  • This can be written as, , noting that the initial charge is CVbat.


Resistance

  • This can be written as, , noting that the initial charge is CVbat.

  • Differentiating gives the current,

  • The voltage across the capacitor is,


Resistance

  • Limiting conditions:

  • At t=0, q= CVbat.

  • At t=inf, q= 0.

q

CVbat

t


Resistance

t

I(t)

Vbat

t


Power energy

Power, Energy


Power

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

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,


Energy1

Energy

  • From the conservation of energy,


Resistance in series and parallel

Resistance in Series and Parallel


Resistance

  • Series:


Resistance

  • From the conservation of energy,


Resistance

  • From the conservation of energy,

  • where,


Resistance

  • From the conservation of energy,

  • where,


Resistance

  • From the conservation of energy,

  • where,


Resistance

  • In general,


Resistance

  • Parallel:


Resistance

  • From the conservation of charge,


Resistance

  • From the conservation of charge,

  • where,


Resistance

  • From the conservation of charge,

  • where,


Resistance

  • From the conservation of charge,

  • where,


Resistance

  • From the conservation of charge,

  • where,


Resistance

  • In general,


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