Electricity & Magnetism
Mathematica Documentation
Mathematica Source Code
Page 2

(mower)

A small magnet may be treated as a single magnetic dipole, of dipole moment M. If the dipole points in the +z direction, its magnetic field will be given by (r is the coordinate perpendicular to the z-axis) Bz = (uo/4pi)M(-r^2+2z^2)/(r^2+z^2)^(5/2) (parallel to the magnetic dipole direction) Br = (uo/4pi)M(3 r z)/(r^2+z^2)^(5/2) (perpendicular to the magnetic dipole direction) Many outboard motors and lawnmowers use rotating magnets passing by coils to create a high voltage for a spark. Instead of having a magnet pass by a coil, we will have a coil pass by a magnet, in order to illustrate the generation of emf. Let the magnet be located at the origin of coordinates, and pointing in the +z direction. We will have the coil traveling with velocity V in the r direction (perpendicular to the z-axis) so that it passes within a distance b of the magnet. The plane of the coil is perpendicular to the z-axis. The coil area A is so small that we may take the magnetic flux through the coil to be the scalar product of the coil area A and the magnetic field. The coil is supposed to be at z=b and y=0 at t=0, so the coordinates of the coil are (z=b, r = Vt). a) Calculate the magnetic flux through the coil as a function of time t. b) Calculate the induced emf through the coil (assuming N turns in the coil) c) Assume M = 2500 A-m^2, V = 120 m/s, and N=2000, b = 5 mm, and a loop area of 0.001 m^2, and plot the induced emf in the coil as a function of t; also plot the magnetic flux in the coil as a function of t.

(noiserc)

We have a signal of interest at 2000 hz, whose amplitude is 100 mV, and an equal, unwanted signal at 60 hz (amplitude 100 mV). You are to build a RC filter to get rid of as much 60 hz as possible, and still leave at least 25 mV of usable signal at 2000 Hz. (The input voltage is applied as the capacitor C and the output voltage is taken at the point where the reisitor R and capacitor C join together). One side of the resistor is connected to the capacitor and the other side is grounded. Consider R values in the range 100 ohms to 5000 ohms, and capacitance values from 0.01 microfarads to 0.5 microfarads. a) Use dsolve to obtain the amplitude A(60) of the 60hz signal at the output of your filter for an arbitrary R and C value. b) Use dsolve to obtain the amplitude A(2000) of the 2000 hz signal at the output of your filter for an arbitrary R and C. c) In the formulas of parts a) and b), do R and C occur separately, or do they occur together? Will separate values of R and C be important, or will a combined value do? d) Make a 3D plot of A(2000) vs R and C in the range given above. e) Make a 3D plot of A(60) vs R and C in the range given above. f) Determine what R and C are needed to give at least a 25 mV signal, with as little noise as possible.

(particle)

A rapidly moving particle of mass M and charge Q travels with constant velocity Vin the x-direction. It passes by a stationary particle of mass M' and charge Q'. The mass M' will be taken as fixed in position and not free to move. The closest distance of approach between the two particles is b.
a) Work out symbolically the net impulse in the y-direction delivered to Q' from Q as it goes by.
For parts b) and c) let M = M' = 1.6 x 10-27 kg, Kinetic energy = 4.1 MeV = 4.1 x 1.6 x10^-13 j, Q = Q' = 1.6x 10^-19C, Kelec = 8.99 x 10^9 N-m^2/C^2.
b) Sum up 100 impulses along the path, from -10 b to +10b.
c) Compare the answers from parts a) and b), and determine the percentage difference. Which one do you think should give the larger answer?

(potplot)

(a) Create a three dimensional contour plot of the electric potential due to three point charges each with magnitude 2.00 µC located at positions (-0.50 m, 0), (+0.50 m, 0), and (0,+0.75 m). The first two charges are positive, the third is negative.
(b) Visually locate any points where a fourth positive test charge would be in equilibrium. State whether the equilibrium is stable or unstable.
(c) Find the magnitude of the electric field at (0, 0.50 m). Going Further (optional):
(d) Calculate the electric field by taking the negative gradient of the potential function.
(e) Make a field plot of unit vectors created from E/|E|.
(f) Make a new vector field by multiplying the unit vector E/|E| by tanh(k*|E|), where k is an adjustable scale constant. Do a 'fieldplot' of this new vector field, with k= 10^5. [You may wish to try some other k values as well.].

(precip)

An electrostatic stack precipitator has a radial electric field along its length, provided by a central wire of radius 0.0200 m, charged with a large negative voltage. The outer radius of the stack is a cylindrical grounded conductor of radius 1.20 m. The electric field at the surface of the central wire is 1.2 x 10^6 V/m, pointing radially inward.
a) Determine the total potential difference applied between the central wire and the grounded outer radius.
b) A particle entering at the bottom of the stack has a mass of 1.4 x 10-15 kg, and picks up a charge of -3.44 x 10-14 C at the central wire. The stack gas velocity is such that the particle will take 5.3 s to travel up the stack. Determine if this particle will be collected at the outer radius of the stack before leaving the stack.
i) Assume the particle does not interact with any other particles.
ii) Assume a spherical particlethe radial motion is impeded by a viscous force given by F(viscous) = -6 pi eta r v, where eta is the kinematic viscosity (2.1 x 10^-5 N-s/m^2), r is the particle's radius and v is the particle's radial velocity. [Obtain the particle's radius by assuming a density of 500 kg/m^3. ]

(rectloop)

A security system has a coil of wire in the shape of a rectangle 1.0 m by 2.0 m. It consists of 500 turns and carries a current of 1 ampere. It may trigger and alarm if the inductance changes when metallic objects change the flux within the loop.
a) Use a 3DPlot to plot the magnetic field in an rectangular region 'interior' to the security coil, 0.8 m by 1.8 m, centered on the larger coil. [This smaller coil has each side parallel to a side of the larger coil and 0.1 m from it.]
b) If we bring a small loop of wire into the 'interior' rectangle, it will change the inductance of the larger loop and may trigger an alarm. What points in the 'interior rectangle' are most sensitive (the greatest flux for a given area of the small loop) and which are least sensitive? [center should be least sensitive, edges most sensitive]

(rodcalc)

A charged rod with a uniform charge density of 2.0 x 10^(-7) C/m extends from -1m to +1m along x-axis.
a) What will the electric potential look like at a distance of 1000 m from the rod? [like that from a point charge; kq/r = 2700/1000 = 2.7 v]
b) Integrate along the rod to obtain the potential at any point (x,y).
c) Evaluate the potential at x=0, y=1000 and compare to your prediction from part a).
d) Evaluate the potential at x=1000, y=2 and compare to part a).
e) From Ey = -dV/dy, obtain a function for Ey at any point (x,y)
f) How should Ey behave very close (0.001 m) to the center of the rod? [Rod looks infinite; Ey = 2 k lambda / r]
g) Evaluate Ey at x=0 and y=1/1000 m. Compare to behavior expected in part f)

(rodint)

A uniformly charged rod, length L and total charge Q.
a) Find the integral which gives the value of the electric potential at an arbitrary point, a distance h beyond the end of the rod, and a distance y perpendicular to the length of the rod. [Assume that the electric potential at infiinity is zero, as usual.]
b) Make a contour plot of the bar for L= 5.0m, Q = 10 microcoulombs. Use the option contours = [40000,60000,80000,100000,120000] to generate five contours of electric potential.

(threept)

Potential and field plots of three point charges
Learning objectives: Translation of standard 1/r potential to off-origin points Physical intuition regarding potential
Identify equilibrium points
Relate potential plot and electric field vectors

(trap)

There is a famous theorem (Earnshaw's theorem) which says in effect that you can't trap a charged particle with a cage made out of only other stationary charges. But it looks like you should be able to build a cage with charges on the 8 corners of a cube, and trap a charge at the center.
The starting material here is the potential on the x-axis due to a charge at (a,b,c) k:=9e9: q:=1e-7: v:=k*q/sqrt(x-a)^2+b^2+c^2):
a) From this, create a function for electric potential due to 4 charges at x=+1m, and y= +/- 1m, z= +/- 1m
b) Plot this function from x=0 to x=2 to show that the positive voltage peaks around x=1, indicating the particle should not be able to escape that way.
c) Now, put the other half of the cube together, 4 charges all at x=-1m, and again at y=+/-1m, and z= +/-1 m. Then plot the total electric potential from x=-2 to x=+2.
All we need is a minimum in the middle in order to trap the particle at the center. How did we do at defeating Earnshaw's theorem?

(wall)

k:=9*10^9;
q:=1*10^(-7);
v:=k*q*Heaviside((a-x)^2-1e-6)/((a-x)^2 + b^2 )^(1/2);
(Heaviside function must be defined)

V due to charge q at (a,b) in xy plane at point (x,0) on the x-axis. The Heaviside (step) function makes the potential vanish very close (0.001 m) to the charge, where it would otherwise start to blow up.

Problem statement. Use the 'function' v in the sample program to create a 'wall' of total charge =100 nC along the x-axis to prevent a proton (q= + 1.6e-19C, m=1.67e-27 kg) from penetrating anywhere along a line from x=0 to x=1 m. The proton is confined to the xz plane, and when far away from the origin it is traveling at 6.9x10^5 m/s. Try a single charge at x=1/2 m and see if that fills the bill, then try two charges of 50 nC each at x=1/4, x=3/4 m.
a) If the proton is to be persuaded not to pass anywhere between x=0 and x=1, what must the minimum value be of the electric potential V between x=0 and x=1? [ Hint: the proton PE = qV. You can find the initial KE of the proton.]
b) Predict qualititatively whether the single or pair of charges should do better, or if it will be a tie. {Cutting q in half gives same v at half the distance, so this would suggest a tie, but it ignores the other charge contributing to the potential. We will do better with 2 charges than one [higher minimum potential]}
c) Do a plot v vs x for a single 'barrier' charge of 100 nC at x=1/2 m. Write down the minimum value of v anywhere between x=0 and x=1.
d) Repeat with two 50 nC charges at 1/4 and 3/4 m to see if minimum v is improved. Report the results. [Can leave it to students to think about actually meeting the spec by moving the 2 charges.]

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