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Lectures 5 and 6 Electrostatics: E-field and Gauss's Law. Introduction
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Gauss's law (which is deduced from Coulomb's law) provides a convenient technique for determining the E-field due to a number different, high-symmetry charge distributions. Furthermore by suitable vector manipulation we arrive at the first of Maxwell's equations.
For a closed surface an outward directed flux is defined as having a positive sign.
where the integral is a surface one over S.
line of length dl has a component dlcos along the arc of the circle
2 radians in a circle
area da makes an angle to the surface of the sphere radius r. The projection of da onto the surface of the sphere is hence dacos
4 steradians in a sphere
Consider a point charge q enclosed by a surface S. Consider a general surface element dS which lies a distance r from q.
The E-field at dS is
and the flux through dS is hence given by
Outward flux across dS
however the term (dScos)/r2 is simply the solid angle d subtended by dS at q (see maths notes for this course). Hence
Outward flux across dS
And total flux across closed surface S is
where the RHS represents the algebraic sum of all charges enclosed by the surface S divided by 0.
Equation (A) gives the integral form of Gauss's law for E-fields. In words it states
'the outward flux of E over any closed surface is equal to the algebraic sum of the charges enclosed by the surface divided by 0.'
What is the electric flux through the following Gaussian surfaces?
A conductor contains at least some charges which are free to move within it. When initially placed in an external E-field the field will penetrate into the conductor and cause the free charges to move (a). However these can only move as far as the surface of the conductor (assuming it is of finite size). Charge collects at the surface and produces an E-field within the conductor which opposes the external field. Equilibrium is quickly reached where the displaced charges produce an internal field which exactly cancels the external one (and hence there is no further movement of charge) (b).
all points of the conductor must be at the same potential.
Solid conductor in (a) carries a net charge. Within conductor E=0 hence flux through Gaussian surface G is zero and hence net charge contained within G is also zero. A solid conductor carries all its excess charge on the surface
spherical but non-point charge.
A conducting sphere of radius a carries a charge Q on its surface. By symmetry the resultant E-field must by spherically symmetric and hence can only have a radial component. Take as a Gaussian surface (G) a sphere of radius r(>a) placed concentric to the conductor.
Flux through G = field x surface area = E.4r2
Total charge enclosed by surface is QE.4r2=Q/0
hence E=Q/(40r2)V= Q/(40r) zero at
An infinite (or very large) sheet carries a uniform charge density . By symmetry the resultant E-field must have a direction normal to the plane and must have the same size at all points a common distance from the plane. Take as a Gaussian surface a cylinder of cross-sectional area A and height 2h.
If field at cylinder ends isEthen total flux is2EA.
Charge enclosed isarea x charge density=A
Hence from Gauss's law2EA=A/0E=/20
This is much easier than the integration method used in Lecture 1.
Find the E-field due to an infinitely long line charge of linear charge density
Find the E-field and electric potential due to a charge of +Q spread out uniformly throughout a sphere of radius a
The integral form is
We now consider the general case where the charge contained within the surface S is continuously distributed with a volume density (which may be depend upon spatial position).
In this case the total charge within Sis given by a suitable volume integral of .
where is the volume enclosed by the surface S.
We now apply Gauss's divergence theorem to convert the surface integral on the LHS to a volume integral.
Gauss's theorem states that 'if a closed surface S encloses a volume then the surface integral of any vector A over S is equal to the volume integral of the divergence of A over .'
Where the surface S encloses the volume
where A is the divergence of the vector A.
Applying the divergence theorem to Gauss's law we have
The two volume integrals can now be combined as they are both evaluated over the same volume
The divergence of a vector A is written as A or div A and is given by
the resultant quantity is a scalar
e.g. if A=3x2yzi+x2z2j+z2k then A=6xyz+2z
Find the divergence of the following cartesian functions (vectors)
Show that the divergence theorem is valid for the vector A=xi+yj+zkand the volume enclosed by a cube of sides length ‘a’ with one corner at the origin and three edges along positive x, y and z.
This is Gauss's law for electric fields in differential form. It is the first of Maxwell's equations.
As the divergence can be thought of as giving the number of field lines starting (if positive) or terminating (if negative) at a given point, the above equation states that ‘the number of field lines starting or terminating at a given point is proportional to the charge density at that point’.
combining these equations we obtain
where in Cartesian co-ordinates
this is Poisson's equation (Laplaces equation if RHS=0)
It was shown in lectures 2 and 3 that the electric potential difference between two points A and B (VBA) is given by
where the integral is a line one between points A and B.
If we now traverse path II in the opposite direction (from B to A) the resultant line integral has a value of -VBA. Hence we can write
This result states that ‘the line integral of E around any closed path in an electrostatic field is zero. Alternatively the work done in taking a test charge round any closed path in an electrostatic field is zero’.
The above result is only true for E-fields produced by static charges (electrostatics). We will see in subsequent lectures that in situations where we have a time varying magnetic field that
Because the circuital law has a null result it is not particularly useful for solving specific problems. However one useful deduction concerns the form of the E-field at the surface of a conductor.
The two ends of the path can be made very small so that they don't contribute to the line integral. Within the conductor we know that E=0. Hence to satisfy the E-field along the top edge of the path must also be zero. Hence there is no tangential component of E just outside the surface of the conductor and hence any E-field must be entirely normal to the surface.
'The E-field just outside a charged conductor is always normal to its surface.'
We have for electrostatics that
Stoke's theorem states that ‘if a closed path L bounds a surface S, then the surface integral of the curl of a vector A over Sis equal to the line integral of Aaround L.’
where xA is the curl of vector A.
as we can make the area S infinitesimally small.
Hence the curl of any electrostatic E-field is always zero. This is the differential form of the circuital law.
This result also follows because E is given by the gradient of the electric potential and it can be shown that the curl of any vector which is derived from the gradient of a scalar function is always zero.
Concept of flux
Gauss’s law in integral formConsequences arising from Gauss’s law – E-fields in conductors.
Use of Gauss’s law to determine the E-field resulting from symmetrical charge distributions
Gauss’s law in differential formPoisson’s and Laplace’s equationsThe circuital law for electrostatic E-fields integral formand differential form E=0