Download Gauss`s Law and Electric Potential Homework

Gauss’s Law and Electric Potential Homework Set
Problem 1. A cube with 1.4 m edges is oriented as shown below in a region of uniform electric
field. Find the electric flux through the right face if the electric field, in newtons/coulomb, is
given by a) 6.00 i b) -2.00 j and c) -3.00 j + 4.00 k d) What is the total flux through the cube for
each of these fields?
z
y
1.40 m
x
Calculate the area of the right face of the cube (at y=1.4 m)
Write the direction of normal to the surface of the right face of the cube (Hint: i, j, or k)
Write an express of the flux Φ as a function of the scalar product of the electric field E
and the area of the normal surface A
For each of electric fields, show the calculation of the electric flux below
Gauss’s Law and Electric Potential Homework Set
For the electric field give in part a) calculate the flux through each face of the cube. (Be sure
to pay attention to the sign of the normal vector i.e. the left side of cube’s normal vector is –i)
Sum the fluxes that were determined in the previous part.
Based on your calculation of the total flux using the E-field in part a), please infer what the total
fluxes using the electric fields of b) and c)
Under what could conditions would the total flux not be the value calculated in the previous
section?
Gauss’s Law and Electric Potential Homework Set
Problem 2 Gauss’ Law for Gravitation is
in which ΦG is the net flux of the gravitational field g though a Gaussian surface that encloses
menclosed. The field g is defined to be the acceleration of a test particle on which exerts menclosed
gravitational force. Derive Newton’s Law of Gravitation from this. What is the significance of
the minus sign?
Assume that the field is created by a uniform spherical shape. In the notes,
is
evaluated. Write that evaluation below.
For the derived expression, substitute E with g and write the new expression below.
Insert the new expression into
and solve for g.
Let menclosed equal the mass of Earth. Derive the gravitational force acting on a particle of
mass m.
What is the significance of the minus sign? Why is there no such explicit sign in the electrical
version of Gauss’ Law?
Gauss’s Law and Electric Potential Homework Set
Problem 3. A long thin-walled cylindrical shell has an inner radius of b and an outer radius of c.
Within the shell a solid charged cylinder of radius a (b>a) is placed in a concentric orientation.
Both have a charge +q distributed uniformly within them. What is the magnitude of the electric
field when a) r<a b) a<r<b, c) b<r<c and d) c<r.
Assume that each cylinder has length L. Calculate the volume for the inner cylinder Vi and
outer cylindrical shell Vs.
Calculate a volume charge density for each object
Choose one of the four Gaussian geometries most appropriate to this problem and write the
evaluation of the surface integral below
For r<a, write the volume of a small cylinder of radius r. Then calculate the total charge
enclosed by this volume by using the appropriate charge density.
Employ the information to this point to calculate the magnitude of the electric field for r<a
using Gauss’ Law
Gauss’s Law and Electric Potential Homework Set
For a<r<b, calculate the charge enclosed in a cylinder with this radius. .
Employ the information to this point to calculate the magnitude of the electric field for
a<r<b using Gauss’ Law.
For b<r<c, write the volume of a cylindrical shell of inner radius b and outer radius r. Then
calculate the total charge enclosed by this volume by using the appropriate charge density. Be
sure to add the charge of the inner cylinder.
Employ the information to this point to calculate the magnitude of the electric field for
a<r<b using Gauss’ Law.
For c<r, write the volume of a cylinder of radius r. Then calculate the total charge enclosed
by this volume by using the appropriate charge density. Be sure to add the charge of the inner
cylinder.
Employ the information to this point to calculate the magnitude of the electric field for c<r
using Gauss’ Law,
Gauss’s Law and Electric Potential Homework Set
Problem 4. In a laboratory experiment, an electron’s weight is just balanced by the force exerted
on the electron by an electric field. If the electric field is due to charges on two parallel,
conducting plates, oppositely charged, separated by 2.3 cm, a) what is the magnitude of the
surface charge density assumed to be uniform on the plates and b) in which direction does the
field point?
Draw a picture of this situation below. Label appropriately. Show the field lines
emanating from each plate. Use two different colors to represent the fields of the two
plates. Let g point down the page.
Draw a free body diagram of the situation.
From the sample problems, write the electric field between two large, charged conducting
plates in terms of the surface charge density σ
Using the free body diagram as guide, write equilibrium equation and solve for σ.
Based on your diagrams, in which direction should the field point?
Gauss’s Law and Electric Potential Homework Set
Problem 5. A solid nonconducting sphere of radius R has a nonuniform charge distribution of
volume charge ρ=ρ0 r/R where r is the distance from the center of the sphere and ρ0 is a constant
of appropriate units. a) Find the total charge on the sphere and b) the electric field inside the
sphere.
Write the expression charge of object in terms of its charge density and its volume
The volume of a sphere of radius a is calculated by taking the surface area of a sphere of
radius r and integrating it over the infinitesimal of dr. The integration limits are between 0
and a. Perform that calculation below.
The total charge on the sphere can be calculated by integrating the volume charge density
multiplied by the surface area of a sphere of radius r over the infinitesimal of dr. In the
case of this problem, the integration limits are between 0 and R. Perform that calculation
below.
Choose the appropriate evaluation of the left hand side of Gauss’ Law for this geometry and
write it below.
Find the total charge with a small sphere of radius r using the methodology discussed
above.
Gauss’s Law and Electric Potential Homework Set
Using the information above, perform algebraic manipulation and solve for the magnitude of
the electric field E.
Gauss’s Law and Electric Potential Homework Set
Problem 6. The figure shows, edge-on, an infinite nonconducting sheet with positive surface
charge density σ on one side.
a) How much work is done by the electric field of the sheet as a small positive test charge q0
is moved from an initial position on the sheet to a final position located a perpendicular
distance z from the sheet?
q0
+
z
++ ++++++++++++++++++++++++++++++++++++
Write the definition of work as a function of the change in potential energy
Write the definition of electric potential as a function of work
Use Gauss’ Law to find the electric field of a charged infinite nonconducting sheet.
Calculate the electric potential for a point charge q0 a distance z from a charged
infinite sheet.
Perform algebraic manipulation to solve the problem.
Gauss’s Law and Electric Potential Homework Set
f
b) Use V f − Vi = − ∫ E ⋅ ds and the result from a) to show that electric potential of an infinite
i
⎞ z , where V is the electric potential at the
sheet can be written as V = V0 − ⎛⎜ σ
0
⎟
ε
2
0⎠
⎝
surface of the sheet.
Perform algebraic manipulation to solve the problem. Hint: Examine the relationship
between electric potential and work.
Gauss’s Law and Electric Potential Homework Set
Problem 7. Consider a point charge q =1.5 x 10-8 C and take V=0 at infinity.
a) What are the shape and dimensions of an equipotential surface having a potential of 30 V
due to q alone?
The equipotential surface of a point charge has the shape most similar to what geometrical
object: sphere, cylinder, or sheet?
Use solution of the electric potential of this geometrical object to solve for the dimensions of
the shape. Hint: there will be only one critical dimension.
b) Are surfaces whose potentials differ by a constant amount (1.0 V, say) evenly spaced?
In the electrical potential equation above, substitute ΔV for V. Perform an analysis of what
happens to the change in dimension. Is it a linear function (which means that the intervals are
evenly spaced)?
Gauss’s Law and Electric Potential Homework Set
Problem 8. In the figure, what is the net potential at point P due to the four point charges, if V=0
at infinity?
+5q +
d
-5q
-5q
d
d
P
d
+5q
+
Write the expression of the electric potential at a distance d from a point charge q
In the space below, show how the superposition principle can be applied to the
electric potentials of a group of point charges Vi to produce a total potential V.
For each charge, write the electric potential that it creates at point P.
Use the superposition principle to calculate the total potential at point P.
Gauss’s Law and Electric Potential Homework Set
Problem 9. The electric potential at points in an xy plane is given by
V= (2.0 V/m2) x2 – (3.0 V/m2) y2 . What are the magnitude and direction of the electric field at
point (3.0 m, 2.0 m)?
Write an expression for the x-component of the electric field in terms of the electric potential.
Take the derivative of V for each component. Hint:
d
y = 0 and vice-versa.
dx
Set each derivative equal to the negative of its respective electric field component.
Evaluate each component at its respective coordinate
Use the Pythagorean Theorem to find the magnitude of the resultant electric field.
Use the trigonometry to find its direction with respect to the +x axis.
Gauss’s Law and Electric Potential Homework Set
Problem 10. In the quark model of fundamental particles, a proton is composed of three quarks:
two “up” quarks, each having charge of +2/3 e and one “down” quark, having charge of -1/3 e.
Suppose the three quarks are equidistant from one another. Take the distance to be 1.32 x 10-15m
and calculate
a) the potential energy of the subsystem of two “up” quarks
Calculate the electric potential from 1 “up” quark at the inter-quark spacing provided
above.
Use this electric potential to calculate the potential energy of “up” “up” subsystem.
b) the total electric potential energy of the three particle system.
Use this electric potential to calculate the potential energy of the “up”-“down”
subsystem.
Use the principle of superposition to calculate the total energy. Hint: U1,2=U2,1 so do
not “double count” energies.