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Parallel Resistors

Introduction

This animation shows the behavior of charge carriers that contribute to the current (charge flow) in two parallel resistors. It takes into account their collisions with the background lattice of the medium which provides the impedance to charge flow. It assumes both the carriers and the background lattice (scatterers) have thermal motion, the former being random translation throughout the resistor and the latter being simple harmonic motion about the nominal position of the nucleus in the lattice.

Voltage Drop

The current, `I`, must equal the Voltage drop, `V`, divided by the resistance, `R`. The current is also defined as the product of the carrier particle charge, `e`, times the carrier density, `n` and the drift speed, `v_d`, times the cross-sectional area of the resistor, `A`.

`I=env_dA`

Carrier Drift Speed

The drift speed is a quantity that requires more physics explanation. First the electric field is

`E=V/L`

where `V` is the voltage accross the resistor and L is the length of the resistor.
An equation for drift speed is

`v_d=eE/m(l/v_("thermal"))`

`e` is the carrier charge, `m` is the carrier mass, `l` is the mean free path, between carrier scatterings and `v_("thermal")` is average the thermal speed of the carriers. As you can see the quantity `eE/m` is the carrier acceleration, a_e,due to the electric field. Also the quantity `tau_e=l/v_("thermal")` is the time between carrier scatterings due to ordinary themal motion. Thus the quantity `v_d=a_etau_e` is the speed gained during a single mean free time and that will be the average drift speed of all of the carriers.

The Mean Free Path

An equation for mean free path,`l`, is obtained by considering the likelihood of scattering in a distance `l` to be 1.0.

`n_(sc)sigmal=1`

where `n_(sc)` is the density of carrier scatterers (often much greater than the density of carriers themselves) and `sigma` is the combined scatterer-carrier scattering cross section which has units of length squared. Then

`l=1/(n_(sc)sigma)`

It's important to note that the probability of electrons scattering from other electrons is much lower than the probability of electrons scattering from the background lattice.

`v_d=eE/mv_("thermal")/(n_(sc)sigma)`

Therefore we can write the current as:

`I=en_cv_dA=en_cE/mv_("thermal")/(n_(sc)sigma)A`

where `N_c` is the total number of electrons in the resistor. It can be seen from this equation that the resistance, R, is

`R=V/I=V/(en_cE/mv_("thermal")/(n_(sc)sigma)A)`

or after removing the `E=V/L` term:

`R=V/I=L/(en_c/mv_("thermal")/(n_(sc)sigma)A)`

Thermal Noise Current

Due to the random thermal motion of the charge carriers, some noise current is generated even when there is no applied voltage. This current can be computed from the number of carriers hitting the top and bottom of the resistor per time increment. The AC variance of the noise current density, `J_("thermal"`,is expected to be:

`deltaJ_("thermal")=en_cv_("thermal")*"random(-1,1)"`

where random(-1,1) is a random number between -1 and +1. Then, since `I=J*A`, the noise current density will be Then the variance of the noise current, `I_("thermal"`,is expected to be:

`deltaI_("thermal")=en_cv_("thermal")A*"random(-1,1)"`

Animation Variables

One variable is the circuit's applied alternating voltage amplitude, `V_(ac)`.

The other variables are the the size of the ions in each of the two resistors. The ion size is effectively the electron scattering cross section which governs the number of electrons hitting the top and bottom of the resistors and thereby contributing to the current.

Animation Graphics

The Applied Voltage

The voltage alternates sinusoidally with minimum and maximum equal to the `Volts` slider value.

The Current

Current is represented by the moving electrons in the circuits as well as the moving electrons in the resistors. The electron speed at any particular time depends on the value of the AC volts furnished by the power supply.

The Ion Size

The ions in the resistors represent impedance to electron movement. Of course, larger ions offer more impedance to electron movement or current.

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