Time-Dependent Simple RC Circuit
The images below are visualizations of the electric charges and
fields in a simple resistor-capacitor circuit. The resistor is
a uniform copper wire joining the two plates. The different
images represent computational steps in the equilibration
of the circuit.
The wires and plates are divided into computational cells, each
a cube 0.5 mm on a side. The entire circuit is 25 mm (about one
inches) across, the wires are 5 mm thick, and this image is a
slice through the mid-plane of the circuit.
The time-step used in the calculations was 0.75 ps, which is
about one-half the time for light to cross a computational cell.
Light would take about 125 ps, or about 165 time steps, to cross
the circuit.
The colors represent the amount of excess charge in each cell,
from red (5000 or more positive elementary charges), to white
(neutral), to blue (5000 or more negative elementary charges).
The arrows show the magnitude and direction of the retarded
electric field due to all the charges calculated at that point.
(The very large electric fields present in and between the plates
are not drawn so the much smaller fields in the wires are visible.)
Click on an image for a larger, clearer picture (725x561 pixels,
about 25k each). Movies with small
(6.1Mb) and large (13.5 Mb) frames are also
available.
| This shows the capacitor at t=0,
when charges are placed on the inner surfaces of the left- and
right-hand plates. The white color indicates no excess
charges are present elsewhere in the circuit. There is no
electric field present in the circuit because no time has elapsed.
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(10 steps = 7.5 ps)
The positive and negative charges are repelling themselves
and spreading outwards from the inner faces of the plates.
The extremely large electric fields near the plates are
not shown.
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(20 steps = 15 ps)
The electric field, made by the newly-introduced
charges, is spreading out at the speed of light. The
electric field pushes charges, and those changes will also
make the electric field. |
|
(40 steps = 30 ps)
We see the finite dipolar electric field from the charges
on the plates of the capacitor. Note that polarization charges
are being formed on the edges of the circuit nearest the plates.
Also note the "surge" of charge moving down the wires.
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(60 steps = 45 ps)
Enough time has passed that the farthest corners of the circuit
are beginning to respond to the charges placed on the capacitor plates.
The "surge" of charge has reached the boundaries of the circuit, and
is piling up there.
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(70 steps = 52 ps)
The electric field in the upper-left and upper-right
corners of the circuit is starting to be affected by the
surfaces charges on the wires.
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(90 steps = 67 ps)
The electric field farther from the upper corners is
beginning to look more like the steady-state configuration
(the electric field vectors will be parallel to the wires
and equal in magnitude throughout the circuit). The
electric field in the lower corners is still farther from
this steady-state configuration, however.
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(150 steps = 112 ps)
These next three pictures show the circuit approaching
steady-state. Note that it takes at least three full
light-crossing times (more than 600 ps) for this to occur.
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(450 steps = 338 ps)
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(850 steps = 638 ps)
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Back to the
time-dependent circuit simulations.
Norris Preyer
