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Link to: physicspages home page. To leave a comment or report an error, please use the auxiliary blog. Reference: Griffiths, David J. This is a brief mathematical interlude since well need the results here when we discuss magnetic dipoles. The vector area of a surface is the integral of the differential area vector over the surface. That is 1 a da S Remember that da is normal to the surface at each point.
As an example, we can calculate a for a hemisphere of radius R. Since the vector area definition is a vector equation, its easiest to split it up into 3 equations in rectangular coordinates. The normal to the hemisphere is in the r direction at every point, and. For a full sphere, we just extend the upper limit on to in the above integrals, and we find that all three components are zero. To see this, we can apply the divergence theorem in the following way.
Thus the vector area of any closed surface is zero. This is because although the actual surface area is non-zero, the vector components always cancel each other out as we integrate over the surface.
Consider now a closed surface and divide it into two parts by cutting it along some closed curve L. Now the shape of the surface is arbitrary, so for some curve L, we can keep the lower surface constant while varying the shape of the upper surface.
That is, alower is held constant while the upper surface varies. However, the equality of the two areas must always hold, so any surface enclosed by L must have the same vector area. This explains why the vector area of a hemisphere is just the area of the circle that defines its base: these are just two different surfaces sharing a boundary curve. We can apply this to get a different formula for a in terms of the boundary curve. Suppose we draw a cone with its vertex at the origin and with its base being the curve L.
The base need not lie in a plane, since L doesnt have to be flat. This wont affect the argument. The vector area of this cone must be the same as any other surface that shares L. The area of this triangle is half the area of the parallelogram with two adjacent sides r and dl, and that in turn is the magnitude r dl. When faced with a continuous distribution of charge, we can work out the electric field as a function of position by using integration instead of summation.
In general, we have. Here r0 is the position of volume element d 3 r0 and r0 is the charge den- sity at that point. There are three types of problems that occur commonly with continuous charge distributions: linear, surface and volume charges. Well do a few examples using linear charges here to see how this works in practice. In many problems in electrostatics, its advisable to make use of any symmetries that the configuration has.
Example 1 Suppose we have a line segment extending along the x axis from L to L. What is the electric field at a point z on the z axis? We can split the problem in two by solving for the x and z components of E separately. Because the z axis divides the linear charge precisely in two, we can use symmetry to conclude that there is no net x component in the field. In the formula above, r r0 is the vector from a point on the linear charge to the point z so we get. The z component of the field will now be half that calculated above:.
By symmetry, the field will be entirely along the z direction, and the contributions from all four sides will be equal. Example 4 Now consider a circular loop of radius r in the xy plane, centred at the origin. By symmetry, the field is entirely in the z direction. A line segment on the circle has length rd , where is the angle in the xy plane. The contribution from all line segments is the same. Now a few examples of surface charge. Example 1 First, we consider a circular disk of radius R with surface charge density lying in the xy plane and centred at the origin.
Find the electric field at a point on the z axis. To solve this we can make use of the solution to the circular loop. In this case were considering a circular ring of circumference 2r and thickness dr, so the amount of charge in the ring is 2r dr and from the earlier solution, the field due to this ring is. This is the field due to an infinite plane of charge. Note that the field is independent of z so is the same no matter how far away from the plane we are.
Example 2 We have a spherical shell of charge with radius R and surface density , centred at the origin. Again, we seek the field at a point on the z axis.
Using spherical coordinates, a point on the sphere has coordinates R, , where is the angle from the positive z axis, and is the azimuthal angle. We can use the cosine law to write the distance between a point on the sphere and the field point:. To get this, we need the angle between r r0 and the z axis. To get this, project the point on the sphere onto the z axis; this gives a point with z coordinate R cos. The remaining distance along the z axis to the field point is therefore z R cos , but this distance is the projection of r r0 onto the z axis.
The cosine of the angle between r r0 and the z axis is this projection divided by r r0 , so we get. Example 3 The result of the last example can be used to find the field due to a sphere that contains a uniform volume charge density.
The field thus increases linearly within the sphere and then falls off as an inverse square outside. Gausss law in electrostatics relates the integral over a closed surface of the electric field to the integral over the enclosed volume of the charge density.
In certain rather specialized situations, Gausss law allows the electric field to be found quite simply, without having to do sometimes horrendous integrals. The situations rely on the geometry of the charge distribution having some kind of symmetry.
Here well give a few examples of how Gausss law can be used in this way. Example 1. We have a spherical shell with radius R and constant surface charge density. Outside the shell E is radially symmetric. The magnitude can be found by integrating E da over the Gaussian surface:. Example 2. Now we take a sphere of radius R that has a uniform volume charge density. Example 3. For an infinitely long charged wire of linear charge density we can choose a cylindrical Gaussian surface of length L and radius s centred on the wire.
By symmetry the field points radially away from the wire and the end caps contribute nothing. Example 4. Inside the sphere, the enclosed charge as a function of r is. In the region a r b we first calculate the enclosed charge.
A coaxial cable has a cylindrical inner core of radius a with uniform volume charge density , and an outer cylindrical shell of radius b with a surface charge density that is of opposite sign to the charge on the core.
The surface charge density is such that the cable is electrically neutral. Inside the inner cylinder, we can use the result of example 3. Since the volume charge density is , the linear charge density for that portion of the cylinder inside radius s is s2 , so the field is. Example 7. An infinite plane slab has thickness 2d, and carries a uniform volume charge density.
Example 8. The two spheres have a region of overlap and we want the electric field within this region. We might be tempted to say that since, in the region of overlap, any vol- ume contains zero net charge since the densities are equal and opposite , there is zero field within this region.
However, the problem with this argu- ment is that when working out the surface integral of the field, there is no obvious symmetry we can invoke. Thus although it is correct to say that any integral E da over a closed surface entirely within the region of overlap is zero, this doesnt automatically translate to the field being zero as it did in earlier examples.
The problem does, however, have a simple solution. If we look back at example 2, we see that the electric field inside a uniformly positively charged sphere is restoring the vector notation. Now suppose that s is the vector from the centre of the negative sphere to the same point. Because the charge is negative, we get.
Thus the field is constant in the region of overlap, although it is not zero. This is a bit of a trick question, since it relies on the field being directly proportional to the radius vector.
For other geometries, no such simple solution exists. An important consequence of the curl of the electric field being zero in electrostatics that is, in the absence of any moving charges comes from Stokess theorem, which for the electric field is. Since the curl is zero, this means that the line integral of the electric field around any closed path is zero. This means that the line integral of E dl along a path connecting two points a and b is independent of the path.
This follows since if the integral did depend on the path, we could choose one path from a to b and then a different path where the integral had a different absolute value on the return path b to a. In that case the integral over the combined path, which is a closed curve, would not be zero, which isnt allowed. In fact the path integral is another way of defining the electric potential function V : the potential difference between two points is the negative of the line integral of E dl along any path connecting those two points:.
Although it is only the potential difference which has any physical signif- icance, it is traditional to define the potential function so that it has a definite value at each point.
Griffiths Introduction to Electrodynamics Textbooks
Estimated delivery business days. Format Hardcover. Condition Brand New. Description This is a re-issued and affordable printing of the widely used undergraduate electrodynamics textbook. This well-known undergraduate electrodynamics textbook is now available in a more affordable printing from Cambridge University Press. The Fourth Edition provides a rigorous, yet clear and accessible treatment of the fundamentals of electromagnetic theory and offers a sound platform for explorations of related applications AC circuits, antennas, transmission lines, plasmas, optics and more. Written keeping in mind the conceptual hurdles typically faced by undergraduate students, this textbook illustrates the theoretical steps with well-chosen examples and careful illustrations.