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Resistance

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Resistance

- 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.

- 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

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

- 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

- 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,

- 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,

- 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,