TimeDependentCurrents

8/7/2019 TimeDependentCurrents

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Time dependent currents

in RC circuits


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A simple RC circuit:

charging a capacitorIntroduce initial conditions:

1. The capacitor is initially uncharged.

2. The switch is closed at t = 0.

Then apply Kirhhoffs

rules to solve for I(t).

This leads to a first order

linear differential

equation for q(t):

- R dq/dt - q/C = 0 => q(t). Then I(t) = dq/dt

t = 0

I(t)


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The solution is:

q(t) = C[ 1 - exp(-t/RC) ]

We define a time constant

= RC (units seconds).

This is a measure of how fast the

circuit reaches equilibrium.

A simple RC circuit

I(t) = Io exp[-t/RC]

with Io = /R


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Example 1

A 5 F capacitor is being charged by a battery. The circuit

resistance is 3 .

How long after closing the switch will it take before thecapacitor reaches 99% of its final voltage?


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Introduce initial conditions:

1. The capacitor is initially charged to Qo2. The switch is closed at t = 0.

As before apply Kirhhoffs

rules to solve for q(t).

q(t) = Qo exp(-t/RC) and I(t) = - dq/dt = Qo/RC exp(-t/RC)


discharging a capacitort = 0

I(t)


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discharging a capacitor

q(t) = Qo exp(-t/RC) I(t) = Qo/RC exp(-t/RC)Io = Qo/RC

= RC


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Measuring DC currentand voltage


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The heart of current, volt and ohm meters is a galvanometerthat can measure very small currents.

The currents pass through a wire

coil in a magnetic field.This causes a torque on the coil

and it rotates to equilibrium opposite

to the torque applied by

an opposing spring.The rotation is recorded on a scale

that is calibrated to the current.

A galvanometer


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Measuring currents (ammeter) and

voltages (voltmeter)

A galvanometer

records small currents

on a digital or analogue meterI

V


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Magnetic forces and fields


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For the electrostatic force we developed a field formalismwhere, given the electric field E, the force on a pointparticle q is:

Fq = q E

We now want to develop a similar field formalism for themagnetic force.

The magnetic force on a pointparticle q is:

Fq = q v x B

Introduction: the magnetic force

SI units of E V/m

SI units of B Tesla


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The magnetic force

on a point particle


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Given the electric and magnetic fields E and B, thecompletely general equation for the force on a

point particle q is:

Fq = q E + q v x B

This is referred to as the Lorentz equation.

The force on an any charged object is obtained by summing(or integrating) over the forces on each point qi (ordifferential dq ) charge component.

The complete EM force

The fundamental

force law of

electromagnetism


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Example 3 (velocity selection)

How can electric and magnetic fields be used to

velocity- select particles?

That is given a beam of particles with charge q and a

continuous spectrum of velocities, select those

with velocity vo .


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Circular motion in a

Uniform B field


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Since a current is moving charges, there is a magnetic forceon a current carrying wire.

For a straight segment of wire carrying a constant current

I, the force isF = I L x B

where L is a vector along the wire in the direction of thecurrent

Magnetic force on

a current carrying wire

TimeDependentCurrents

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