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Electromagnetism

Uniform Spherical Charge Distribution Suppose we have a point charge outside a spherical charge distribution of radius and charge

density .

By Coloumb's Law:

Since :

From the diagram:

Hence the integral becomes:

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Since by symmetry and will be zero, we need only worry about the z-component of :

Hence:

Or because we know it's in the direction we can just worry about the magnitude of the field:

For simplicity, convert to spherical coordinates:

Nothing in the integrand depends on , so it will integrate out simply:

Let

Use mathematica to find this integral:

Simplify:

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The total charge for a homogenous sphere of charge density is simply:

Hence:

Substituting this into we find:

Which is simply Coloumb's law for two point charges

Infinite Flat Sheet of Charge What is the electric field at distance P from a uniform, infinite, flat sheet of electric charge?

From the diagram:

By Coloumb's Law:

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Since :

By symmetry we know that , hence only concern ourselves with

From integral tables:

Equations for Scalar Potential

Dipole Potential

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The potential at point P is given by:

Using the cosine rule we can express and in terms of and :

Using the binomial theorem:

Keeping only the first two terms of

yields:

Likewise for

Substituting these expressions into our equation for potential:

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The term in the numerator is called the dipole moment

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Method of Images The method of images is a handy tool that we can use to solve for the potential, electric field, etc, in

situations where using Maxwell's equations directly would be difficult. Basically the idea is simply to

add a mirror image of the charge or charges in question in an area that is outside the boundary

conditions of our problem. The method of image charges is based on the uniqueness theorem, which

states that any electric field that satisfies a given set of boundary conditions is unique; it is the only

electric field that can satisfy those conditions. Thus, if we can find a system of charges that satisfies

the same boundary conditions as our original problem, then we know that this solution must be the

same as the solution to our original problem.

We can solve this problem using the method of images, by placing a fictitious charge inside the

sphere (hence outside the boundary conditions).

Given the boundary conditions, the potential on the sphere edge must be zero.

Rearranging:

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Combining the two equations:

Hence substituting in:

But does this image satisfy the boundary conditions at other points, such as P3?

Using the cosine rule we know that:

Substituting in:

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This tells us that the image method we have used is valid. Hence the potential at an arbitrary angle

and distance is given by:

To find the electric field simply use the equation in spherical coordinates.

Bio-Savat Law

Use cartesian coordinates:

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Hence we can find the cross product as:

Substituting this into the Biot-Savart Law:

Magnetic Dipole

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Vector potential is given by:

In a loop about the dipole:

We also need the distance :

Through some trig magic (see diagram), it turns out that

If ,

, hence:

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Substituting in our and into the expression for :

By symmetry the y-component will be zero.

Apparently this is the same as:

The magnetic field is given by:

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Point Charge in Dielectric

First find D using Gauss' law for materials:

Given the general formula for D:

Outside the material, , hence:

Outside the material,

, hence:

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To find P

Parallel Plate Capacitor with Dielectric

From the question the expression for is given by:

To find :

Given the generic expression for :

We find the electric field to be:

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The potential difference is given by:

Parallel Plate Capacitor with Partly Inserted Dielectric

Begin with the generic form of Gauss' Law for materials:

With dielectric present:

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With no dielectric present:

Hence total charge is given by:

From the definition of capacitance:

Magnetic Materials

Diamagnetism Diamagnetism is the property of an object or material that causes it to create a magnetic field in

opposition to an externally applied magnetic field. It is a quantum mechanical effect that occurs in

all materials; where it is the only contribution to the magnetism the material is called a diamagnet.

However, for materials that show some other form of magnetism (such as ferromagnetism or

paramagnetism), the diamagnetic contribution becomes negligible. Materials called diamagnetic are

those that non-physicists generally think of as non-magnetic, and include water, wood, most organic

compounds such as petroleum and some plastics, and many metals including copper, particularly the

heavy ones with many core electrons, such as mercury, gold and bismuth.

Diamagnetism occurs at the quantum level, as a result of altered electron orbit motions when an

external magnetic field is applied. By Lenz's law, the induced magnetic field always is in opposite

direction to the applied magnetic field, causing repulsion.

Paramagnetism Paramagnetism is a form of magnetism whereby certain materials are attracted by an externally

applied magnetic field. Constituent atoms or molecules of paramagnetic materials have permanent

magnetic moments (dipoles), even in the absence of an applied field. The permanent moment

generally is due to the spin of unpaired electrons in atomic or molecular electron orbitals (see

Magnetic moment). In pure paramagnetism, the dipoles do not interact with one another and are

randomly oriented in the absence of an external field due to thermal agitation, resulting in zero net

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magnetic moment. When a magnetic field is applied, the dipoles will tend to align with the applied

field, resulting in a net magnetic moment in the direction of the applied field.

Ferromagnetism Permanent magnets (materials that can be magnetized by an external magnetic field and remain

magnetized after the external field is removed) are either ferromagnetic or ferrimagnetic, as are

other materials that are noticeably attracted to them. Only a few substances are ferromagnetic. The

common ones are iron, nickel, cobalt and most of their alloys, some compounds of rare earth metals,

and a few naturally-occurring minerals such as lodestone. Ferromagnetism is a property not just of

the chemical make-up of a material, but of its crystalline structure and microscopic organization.

In materials with a filled electron shell, the total dipole moment of the electrons is zero because the

spins are in up/down pairs (these act as diamagnets). Only atoms with partially filled shells (i.e.,

unpaired spins) can have a net magnetic moment, so ferromagnetism only occurs in materials with

partially filled shells. These unpaired dipoles tend to align in parallel to an external magnetic field, an

effect called paramagnetism.

Ferromagnetism involves an additional phenomenon, however: The dipoles tend to align

spontaneously, giving rise to a spontaneous magnetization, even when there is no applied field.

Regions of aligned magnetic dipole moment are called domains. Ordinarily, ferromagnetic materials

have domains with alignments in random directions, so overall the object has no net magnetic

moment. However, when placed in a suitably strong magnetic field, the domains will spontaneously

align, and remain aligned (or mostly aligned) after the external magnetic field is removed. The

resulting permanent magnet will remain magnetised.

Boundary Conditions for B and H

It is known that for almost all conductors:

Hence:

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Combining these two equations:

Solving for the magnitude of :

Surface Interface Equations Note that surface currents only exist in superconductors, hence almost always

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Magnetic Field in Toroid with Gap

From Gauss' law for magnetism:

Consider a Gaussian surface shaped like a cylinder extending either side of the edge of the torroid.

We can thus split this surface integral into two parts, one over the inner flat surface of the cylinder

(inside the torroid), and one over the outer flat surface (in the gap):

The normal vectors for and point in opposite directions, but are otherwise identical, so:

From the interface continuity equations we also know that:

Since the field is normal to the interface between the media at all points, we can simplify this to:

By Ampere's law we have:

Considering an Amperian loop around the torroid:

Since and :

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Hence the magnetic field strength in the torroid is given by:

The Continuity Equation We begin with the basic equation for current:

This can be written in terms of an integral over the surface current density:

Writing the total charge as an integral over the charge volume density , and applying the

divergence theorem to the right hand side:

Since this equation must hold for any arbitrary volume, we can drop the integrals:

In words, this states that the loss of charge density from a point is equal to the current density

flowing from that point.

The Wave Equation Begin with Faraday's Law:

Take the divergence of both sides:

Use the following vector identity:

Hence:

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By Gauss' Law and Ampere's Law:

Substituting these equations in:

If current and charges are absent this becomes:

This is the wave equation, which indicates that electromagnetic waves can propagate themselves in

a vacuum, travelling that the speed of light

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Energy Density of Fields Hence the energy density for the electric field is given by:

Similarly for the energy density of the magnetic field is:

The total energy density is simply the sum of the electric and magnetic components:

For monochromatic plane waves, we know that:

Substituting in this expression:

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This tells us that for electromagnetic plane waves, the magnetic energy density must be equal to the

electric energy density.

This energy density, however, only applies to a particular infinitesimal unit of time. The actual

measured intensity will be determined by an average of many such units of time, and is given by:

The term

is the reciprocal of the impedance , which measures the resistance of the given

medium to the propagation of electromagnetic waves.

Poynting Vector The Poynting Vector represents the power flow per unit area in the direction of propagation. It is

simply the wave intensity multiplied by the unit normal vector:

For plane waves, , hence:

Since and are perpendicular:

We can also show that the Poynting Vector satisfies the continuity equation:

Considering the sources of power, we take the divergence:

Using a vector identity:

In the absence of any charges or currents:

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Substituting these in:

So long as the medium is constant and isotropic:

From our definition above, the term in brackets is equal to the energy density :

This is simply the continuity equation: the divergence of a vector field (in this case ), is equal to the

negative of the rate of change of the density of some quantity (in this case energy) with respect to

time.

Absorption in Conductors In conducting materials, the current term is not zero, but is given by . Hence we derived a

slightly different form of the wave equation:

This last term was absent in the previous equation, as in a vacuum obviously . This additional

term means that the differential equation now has complex plane-wave solutions:

The imaginary part of causes the wave to decay; that is its intensity is attenuated over distance.

The distance that it takes before the amplitude is reduced by a factor of is called the skin depth:

Where depends on the material and the frequency of oscillation.

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Electric Field Intensity Example

Using the general formula for :

The phase of this wave is given by . At the point of origin this will be equal to zero (as

), hence we have: