GAUSS’S LAW
To simplify the electric field calculations using calculus
SYMMETRY PROPERTIES
Many physical systems have symmetry; for example, a cylindrical body doesn’t
look any difference after you’ve rotated it around its axis and a charged metal
sphere looks just the SAME after you turned it about any axis through its axis.
IF WE PLACE THE CHILD INSIDE THE CHAMBER
THE ELECTRIC FIELD INSIDE THE CAVITY WITHIN A CONDUCTOR IS ZERO,
SO THERE WOULD BE NO ELECTRIC EFFECT ON THE CHILD
GAUSS’S LAW
Given any general distribution of charge, we surround it with an imaginary surface
that enclose the charge. Then we look at the electric field at various points on this
imaginary surface. Gauss’s Law is a relationship between the field at all the points
on the surface and the total charge enclosed with the surface.
This may sound like an indirect way of expressing things, but it turns out that to be
tremendously useful relationship.
Review on Electric Field E = Fo/Qo
ELECTRIC FLUX ( FLOW )
THE MEASURE OF THE ‘FLOW” OF ELECTRIC FIELD THROUGH A SURFACE.
IT IS EQUAL TO THE PRODUCT OF AN AREA AND THE PERPENDICULAR
COMPONENT OF ELECTRIC FIELD OVER THE SURFACE.
ΦE = ∫ E cosϕ dA
= ∫ E⟂dA = ∫ E . dA
GAUSS’S LAW
STATES THAT THE TOTAL ELECTRIC FLUX THROUGH A CLOSED SURFACE,
WHICH CAN BE WRITTEN AS THE SURFACE INTEGRAL OF THE COMPONENT OF
E NORMAL TO THE ENCLOSED SURFACE, EQUAL A CONSTANT TIMES THE TOTAL
CHARGE Q BY THE SURFACE.
IT IS LOGICALLY EQUIVALENT TO COULOMB’S LAW, BUT IT’S USE GREATLY
SIMPLIFIES PROBLEM WITH HIGH DEGREE OF SYMMETRY.
WHEN EXCESS CHARGE IS PLACED ON A SOLID CONDUCTOR AND IS AT REST, IT
RESIDES ENTIRELY ON THE SURFACE AND ELECTRIC FIELD IS ZERO
EVERYWHERE IN THE MATERIAL OF THE CONDUCTOR
SAMPLE PROBLEM #1
ELECTRIC FLUX THROUGH A DISK
QUALITATIVE APPROACH
1.
How many net electric field lines would pass through a Gaussian surface
located totally within the a set of oppositely charged parallel plates?
ANSWER:
ZERO, since the number of lines going into the surface equals the number of
lines coming out of the surface.
2. A Gaussian surface surrounds an object with a net charge of +5.0µC. Which of
the following is true?
a.
b.
c.
More electric field lines will point outward than inward
More electric field lines will point inward than outward
The net number of field lines through the surface is zero
ANSWER:
A.
Since the field lines point away from positive charge.
3. Two concentric spherical surfaces enclose a point charge. The radius of the
outer sphere is twice that of the inner one. Which sphere will have more electric
field lines penetrating?
ANSWER
The SAME since both enclose the charge.
4. The same Gaussian surface is used to surround two charged objects
separately. The net number of electric field lines penetrating the surface is the
same in both cases, but they are oppositely directed. What can you say about eh
net charge on the two objects?
ANSWER
Since the number of lines is proportional to the charge, the net charges are
EQUAL AND OPPOSITE IN SIGN.
5. If a Gaussian surface has 16 field lines leaving it when it surround a point
charge of + 10µC and 75 field lines entering it when it surrounding an unknown
point charge, what is the amount of charge on the unknown?
ANSWER:
q=75/16 ( +10µ C )
= + 46.9 µ C
6. If 10 field lines leave a Gaussian surface when it completely surrounds the
positive end of an electric dipole, what is the count if it surrounds just the other
end?
ANSWER
A dipole consists of a pair of equal and opposite charges.
So there will be 10 FIELD LINE ENTERING ( NEGATIVE )
7. Suppose a Gaussian surface encloses both positive point charge that has 6
field lines leaving it and a negative point charge with twice the magnitude of the
charge as the positive one. What is the number of field lines passing through the
Gaussian surface?
ANSWER
The net of field line is +6 + ( 2x-6) = -6 or 6 LINES ENTERING
8. If the net number of electric field lines point out of a Gaussian surface, does that
necessarily mean there are no negative charges in the interior?
ANSWER:
NO. All it means is that there is more positive charge than negative. A special case
would be no negative charge and only positive charge.