FORMULAE
Electric Field: ܧሬԦ =
ଵ
ଵ
ଵ
ௗ
σ ݎෝ; ܧሬԦ =
ݎƸ ; ܧሬԦ =
ݎ Ƹ .
ସగఌ మ
ସగఌ మ ప
ସగఌ మ
Electric Force: ܨԦ = ݍ ܧሬԦ
Electric Dipole: ߬Ԧ = Ԧ × ܧሬԦ
Gauss Law for electrostatics: ܧ ׯሬԦ ή ݀ܣԦ =
ொ
ఌ
Electric Potential: οܸ = ܸ െ ܸ = െ ܧሬԦ ή ݀ݏԦ; Energy: οܷ = ݍοܸ; Work: ܹ = െοܷ
ܸ=
ଵ
ଵ
ଵ
ௗ
σ ; ܸ =
;ܸ=
ସగఌ
ସగఌ
ସగఌ
ሬԦܸ = െ డ ݔො െ డ ݕො െ డ ݖƸ
ܧሬԦ = െ
డ௫
Capacitors: = ܥ
డ௬
డ௭
ஂ ఌ
ொ
; ; = ܥparallel: ܥ = ܥଵ + ܥଶ + ;ڮ
ௗ
ଵ
ଶ
series:
ଵ
ଵ
ଵ
= + +ڮ
భ
మ
ଵ
ଶ
Energy density: ܧߝ = ݑଶ
Energy: ܷ = ܸܳ;
ANTIDERIVATIVES
ௗ௬
ඥ మ ା௬ మ
= ln( ݕ+ ඥ ݕଶ + ܽଶ );
ଵ ( మ ା௬ మ )శభ
,
ାଵ
ܽ(ݕ ଶ + ݕଶ ) ݀ = ݕଶ
ௗ௬
( మ ା௬మ )య/మ =
௬
మ ඥ(మ ା௬ మ )
;
݊ ് െ1
2
YOU MUST SHOW ALL STEPS TO GET FULL CREDIT. THIS APPLIES TO BOTH
MULTIPLE CHOICE AND LONG PROBLEMS.
IF USING EXTRA SHEETS, CLEARLY LABEL YOUR WORK TO INDICATE TO
WHICH PROBLEM IT BELONGS.
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[1] A spherical object of mass m and charge +q is placed a distance d above a large,
uniformly charged, nonconducting sheet with a surface charge density –ߪ, as shown in
the figure. Acceleration due to gravity ݃റ is non-negligible and is pointing downward. In
this problem, d is much
smaller
than
the
dimensions of the sheet
such that the magnitude of
the electric field due to the
charged sheet is uniform at
the object’s location and
given by ߪ/2ߝ.
Calculate the number of electrons that need to be added to or removed from the object
such that the object remains at the location shown (below, ݁ is the magnitude of charge
on an electron).
(A)
+
ଶఌబ
ఙ
ଶఌబ
ఙ
(B) െ +
(C)
ఙ
ସగఌబ ௗ మ
(D)
ఌబ
ఙ
(E)
ఙ
ଶఌబ
[2] Work required to double the separation between two point charges + ݍand – ݍinitially
distance ݎapart, is best given by
(A)
మ
݇ݍ2
(B) 2 ݎ
(C)
మ
ଶ
(D) െ2
మ
(E)
మ
ସ మ
[3] An electron released from rest at a location where V=0 but ് ܧ0 will
(A) not move since it has no initial velocity
(B) move at a constant speed along the V=0 line
(C) accelerate along the direction of decreasing V
(D) accelerate along the direction of increasing V
(E) not move since it was placed at a location where V=0
[4] A horizontal rod of length L has one end at the origin and the other end on the
positive x-axis. The linear charge density, ߣ, in the rod varies along its length and is
given by ߣ( = )ݔെ(ܭ
௫మ
constant):
ଶ
(A) െܮܭଷ (B) െ ଷ ܮܭଶ
െ )ܮ. The total charge on the rod is (assume K is a positive
ଶ
(C) ଷ ܮܭଶ
ଵ
(D) െ ଷ ܮܭଷ
(E) 0
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[5] Three point charges are on the x axis. Charge Q1 is at = ݔ0, Q2 is at ܮ = ݔ, and
Q3 is at ܮ = ݔ/3. If the total force on Q3 due to Q1 and Q2 is zero, the ratio Q1/ Q2
equals
(A) -9/4
(B) -1/4
(C) 1/2
(D) 1/4
(E) 1
[6] Three capacitors C1, C2, and C3 are
connected to a battery as shown. The three
capacitors have equal capacitances. Which
capacitor stores the most potential energy?
(A) All three store the same amount of energy
(B) C2 or C3
(C) C3
(D) C2
(E) C1
[7] Two spherical charged conductors placed some distance apart produce the
equipotential surfaces shown in the figure. The value of each potential surface is also
marked, with the 0 V line being equidistant from the two charges.
What is the best comparison of the electric fields at points A and F?
(A) Both point to the left
(B) Both point to the right
(C) Field at A points to the right and field at F points to the left
(D) Field at A points to the left and field at F points to the right
(E) Cannot know without knowing nature of the charges
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[8] If
and
are nonzero vectors for which
(A)
×
=0
(B)
is parallel to
(C) |
×
| = AB
(D) |
×
|=1
Â
(E) |
= 0, it must follow that
×
ෝ + ܣ௬ ܤ௬ ࢟
ෝ + ܣ௭ ܤ௭ ࢠො
| = ܣ௫ ܤ௫ ࢞
[9] Two very large parallel sheets a distance d apart have their centers directly opposite
each other. The sheets carry equal but opposite uniform surface charge densities. A
point charge that is placed near the middle of the sheets a distance d/2 from each of
them feels an electrical force F due to the sheets. If this charge is now moved closer to
one of the sheets so that it is a distance d/4 from that sheet, what force will it feel?
(A) 4F
(B) 2F
(C) F
(D) F/2
(E) F/4
[10] The graph in the figure shows the electric field strength (not the field lines) as a
function of distance from the center for a pair of concentric uniformly charged spheres.
Which of the following situations could the graph plausibly represent?
(A) a positively charged conducting sphere within another positively charged conducting
sphere
(B) a positively charged conducting sphere within an uncharged conducting sphere
(C) a solid nonconducting sphere, uniformly charged throughout its volume, inside of a
positively charged conducting sphere
(D) a positively charged nonconducting thin-walled spherical shell inside of a negatively
charged conducting sphere
(E) a positively charged nonconducting thin-walled spherical shell inside of another
positively charged nonconducting thin-walled spherical shell
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[11] The figure shows an annulus with inner radius
ݎ and outer radius 4ݎ and a uniform surface
charge density +ߪ. We are interested in point P
that lies along the central axis of the annulus and
is a distance ݖvertically above the center.
(A) Calculate the electric field ܧሬറ at point P. Be sure
to indicate the field direction and magnitude.
(B) By assuming the potential V=0 at infinity,
calculate the electric potential at point P.
(C) How much work is done in moving an electron
with charge െ݁ from = ݖ2ݎ to = ݖ0?
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[12] An insulating hollow sphere of inner radius ܴଵ and outer radius ܴଶ carries charge
described by the volume charge density ߩ(ߙ = )ݎ/ݎ. Here ߙ is a positive constant.
ሬറ as a
(A) Calculate the magnitude and direction of ܧ
function of ݎin the region ܴଵ < ܴ < ݎଶ . You may find
Gauss’s law useful in this context.
ሬሬሬറ
ሬറ . ݈݀
(B) Using the relationship ܸ െ ܸ = െ ܧ
or
otherwise, determine the electrostatic potential ܸ as a
function of ݎin the region ܴଵ < ܴ < ݎଶ . You can assume
ܸ (ܴଵ ) = 0.
(C) Imagine now that you have a choice of point charge ݍthat you can place at
the center of the sphere such that the net ܧሬറ becomes a constant and equal to
ߙ/2ߝ in the region ܴଵ < ܴ < ݎଶ . Determine the sign and magnitude of ݍthat you
ሬറ a constant.
would pick to make ܧ
9
10
11
0
