Electrostatics Free Response Practice Questions | AP Physics C Electricity and Magnetism - Grade 9 PDF Download

 Page 1


Section II: Free Response
1. In the figure shown, all four charges (+ Q, + Q, - q, and - q) are situated at the corners of a
square. The net electric force on each charge + Q is zero.
(a) Express the magnitude of q in terms of Q.
(b) Is the net electric force on each charge - q also equal to zero? Justify your answer.
(c) Determine the electric field at the center of the square.
2. Two charges, + Q and +2 Q, are fixed in place along the y-axis of an x-y coordinate system as
shown in the figure below. Charge 1 is at the point (0, a), and Charge 2 is at the point (0, -2 a).
(a) Find the electric force (magnitude and direction) felt by Charge 1 due to Charge 2.
(b) Find the electric field (magnitude and direction) at the origin created by both Charges 1 and
2.
(c) Is there a point on the x-axis where the total electric field is zero? If so, where? If not,
explain briefly.
(d) Is there a point on the y-axis where the total electric field is zero? If so, where? If not,
explain briefly.
Page 2


Section II: Free Response
1. In the figure shown, all four charges (+ Q, + Q, - q, and - q) are situated at the corners of a
square. The net electric force on each charge + Q is zero.
(a) Express the magnitude of q in terms of Q.
(b) Is the net electric force on each charge - q also equal to zero? Justify your answer.
(c) Determine the electric field at the center of the square.
2. Two charges, + Q and +2 Q, are fixed in place along the y-axis of an x-y coordinate system as
shown in the figure below. Charge 1 is at the point (0, a), and Charge 2 is at the point (0, -2 a).
(a) Find the electric force (magnitude and direction) felt by Charge 1 due to Charge 2.
(b) Find the electric field (magnitude and direction) at the origin created by both Charges 1 and
2.
(c) Is there a point on the x-axis where the total electric field is zero? If so, where? If not,
explain briefly.
(d) Is there a point on the y-axis where the total electric field is zero? If so, where? If not,
explain briefly.
(e) If a small negative charge, - q, of mass m were placed at the origin, determine its initial
acceleration (magnitude and direction).
3. A conducting spherical shell of inner radius a and outer radius b is inside (and concentric
with) a larger conducting spherical shell of inner radius c and outer radius d. The inner shell
carries a net charge of +2 q, and the outer shell has a net charge of +3 q.
(a) Determine the electric field for
(i) r < a
(ii) a < r < b
(iii) b < r < c
(iv) c < r < d
(v) r > d
(b) Show in the figure the charges that reside on or inside each of the two shells.
4. A positively charged, thin nonconducting rod of length l lies along the y-axis with its midpoint
at the origin. The linear charge density within the rod is uniform and denoted by ?.
Points P
1
 and P
2
 lie on the positive x-axis, at distances x
1
 and x
2
, respectively from the rod.
Page 3


Section II: Free Response
1. In the figure shown, all four charges (+ Q, + Q, - q, and - q) are situated at the corners of a
square. The net electric force on each charge + Q is zero.
(a) Express the magnitude of q in terms of Q.
(b) Is the net electric force on each charge - q also equal to zero? Justify your answer.
(c) Determine the electric field at the center of the square.
2. Two charges, + Q and +2 Q, are fixed in place along the y-axis of an x-y coordinate system as
shown in the figure below. Charge 1 is at the point (0, a), and Charge 2 is at the point (0, -2 a).
(a) Find the electric force (magnitude and direction) felt by Charge 1 due to Charge 2.
(b) Find the electric field (magnitude and direction) at the origin created by both Charges 1 and
2.
(c) Is there a point on the x-axis where the total electric field is zero? If so, where? If not,
explain briefly.
(d) Is there a point on the y-axis where the total electric field is zero? If so, where? If not,
explain briefly.
(e) If a small negative charge, - q, of mass m were placed at the origin, determine its initial
acceleration (magnitude and direction).
3. A conducting spherical shell of inner radius a and outer radius b is inside (and concentric
with) a larger conducting spherical shell of inner radius c and outer radius d. The inner shell
carries a net charge of +2 q, and the outer shell has a net charge of +3 q.
(a) Determine the electric field for
(i) r < a
(ii) a < r < b
(iii) b < r < c
(iv) c < r < d
(v) r > d
(b) Show in the figure the charges that reside on or inside each of the two shells.
4. A positively charged, thin nonconducting rod of length l lies along the y-axis with its midpoint
at the origin. The linear charge density within the rod is uniform and denoted by ?.
Points P
1
 and P
2
 lie on the positive x-axis, at distances x
1
 and x
2
, respectively from the rod.
(a) Use Gauss’s law to approximate the electric field at point P
1
, given that x
1
 is very small
compared to l. Write your answer in terms of ?, x
1
, and fundamental constants.
(b) What is the total charge Q on the rod?
(c) Compute the electric field at point P
2
, given that x
2
 is not small compared to l. For x
2
 = l,
write your answer in terms of Q, l, and fundamental constants. You may use the fact that
5. A solid glass sphere of radius a contains excess charge distributed throughout its volume such
that the volume charge density depends on the distance r from the sphere’s center according
to the equation
?( r) = ?
s
(r/ a)
where ?
s
 is a constant.
(a) What are the units of ?
s
?
(b) Compute the total charge Q on the sphere.
(c) Determine the magnitude of the electric field for
(i) r < a
(ii) r = a
Write your answers to both (i) and (ii) in terms of Q, a, r, and fundamental constants.
(d) Sketch the electric field magnitude E as a function of r on the graph below. Be sure to
indicate on the vertical axis the value of E at r = a.
Page 4


Section II: Free Response
1. In the figure shown, all four charges (+ Q, + Q, - q, and - q) are situated at the corners of a
square. The net electric force on each charge + Q is zero.
(a) Express the magnitude of q in terms of Q.
(b) Is the net electric force on each charge - q also equal to zero? Justify your answer.
(c) Determine the electric field at the center of the square.
2. Two charges, + Q and +2 Q, are fixed in place along the y-axis of an x-y coordinate system as
shown in the figure below. Charge 1 is at the point (0, a), and Charge 2 is at the point (0, -2 a).
(a) Find the electric force (magnitude and direction) felt by Charge 1 due to Charge 2.
(b) Find the electric field (magnitude and direction) at the origin created by both Charges 1 and
2.
(c) Is there a point on the x-axis where the total electric field is zero? If so, where? If not,
explain briefly.
(d) Is there a point on the y-axis where the total electric field is zero? If so, where? If not,
explain briefly.
(e) If a small negative charge, - q, of mass m were placed at the origin, determine its initial
acceleration (magnitude and direction).
3. A conducting spherical shell of inner radius a and outer radius b is inside (and concentric
with) a larger conducting spherical shell of inner radius c and outer radius d. The inner shell
carries a net charge of +2 q, and the outer shell has a net charge of +3 q.
(a) Determine the electric field for
(i) r < a
(ii) a < r < b
(iii) b < r < c
(iv) c < r < d
(v) r > d
(b) Show in the figure the charges that reside on or inside each of the two shells.
4. A positively charged, thin nonconducting rod of length l lies along the y-axis with its midpoint
at the origin. The linear charge density within the rod is uniform and denoted by ?.
Points P
1
 and P
2
 lie on the positive x-axis, at distances x
1
 and x
2
, respectively from the rod.
(a) Use Gauss’s law to approximate the electric field at point P
1
, given that x
1
 is very small
compared to l. Write your answer in terms of ?, x
1
, and fundamental constants.
(b) What is the total charge Q on the rod?
(c) Compute the electric field at point P
2
, given that x
2
 is not small compared to l. For x
2
 = l,
write your answer in terms of Q, l, and fundamental constants. You may use the fact that
5. A solid glass sphere of radius a contains excess charge distributed throughout its volume such
that the volume charge density depends on the distance r from the sphere’s center according
to the equation
?( r) = ?
s
(r/ a)
where ?
s
 is a constant.
(a) What are the units of ?
s
?
(b) Compute the total charge Q on the sphere.
(c) Determine the magnitude of the electric field for
(i) r < a
(ii) r = a
Write your answers to both (i) and (ii) in terms of Q, a, r, and fundamental constants.
(d) Sketch the electric field magnitude E as a function of r on the graph below. Be sure to
indicate on the vertical axis the value of E at r = a.
Page 5


Section II: Free Response
1. In the figure shown, all four charges (+ Q, + Q, - q, and - q) are situated at the corners of a
square. The net electric force on each charge + Q is zero.
(a) Express the magnitude of q in terms of Q.
(b) Is the net electric force on each charge - q also equal to zero? Justify your answer.
(c) Determine the electric field at the center of the square.
2. Two charges, + Q and +2 Q, are fixed in place along the y-axis of an x-y coordinate system as
shown in the figure below. Charge 1 is at the point (0, a), and Charge 2 is at the point (0, -2 a).
(a) Find the electric force (magnitude and direction) felt by Charge 1 due to Charge 2.
(b) Find the electric field (magnitude and direction) at the origin created by both Charges 1 and
2.
(c) Is there a point on the x-axis where the total electric field is zero? If so, where? If not,
explain briefly.
(d) Is there a point on the y-axis where the total electric field is zero? If so, where? If not,
explain briefly.
(e) If a small negative charge, - q, of mass m were placed at the origin, determine its initial
acceleration (magnitude and direction).
3. A conducting spherical shell of inner radius a and outer radius b is inside (and concentric
with) a larger conducting spherical shell of inner radius c and outer radius d. The inner shell
carries a net charge of +2 q, and the outer shell has a net charge of +3 q.
(a) Determine the electric field for
(i) r < a
(ii) a < r < b
(iii) b < r < c
(iv) c < r < d
(v) r > d
(b) Show in the figure the charges that reside on or inside each of the two shells.
4. A positively charged, thin nonconducting rod of length l lies along the y-axis with its midpoint
at the origin. The linear charge density within the rod is uniform and denoted by ?.
Points P
1
 and P
2
 lie on the positive x-axis, at distances x
1
 and x
2
, respectively from the rod.
(a) Use Gauss’s law to approximate the electric field at point P
1
, given that x
1
 is very small
compared to l. Write your answer in terms of ?, x
1
, and fundamental constants.
(b) What is the total charge Q on the rod?
(c) Compute the electric field at point P
2
, given that x
2
 is not small compared to l. For x
2
 = l,
write your answer in terms of Q, l, and fundamental constants. You may use the fact that
5. A solid glass sphere of radius a contains excess charge distributed throughout its volume such
that the volume charge density depends on the distance r from the sphere’s center according
to the equation
?( r) = ?
s
(r/ a)
where ?
s
 is a constant.
(a) What are the units of ?
s
?
(b) Compute the total charge Q on the sphere.
(c) Determine the magnitude of the electric field for
(i) r < a
(ii) r = a
Write your answers to both (i) and (ii) in terms of Q, a, r, and fundamental constants.
(d) Sketch the electric field magnitude E as a function of r on the graph below. Be sure to
indicate on the vertical axis the value of E at r = a.
Section II: Free Response
1. (a) From the figure below, we have F
1-2
 = F
1
/cos 45°.
Since the net force on + Q is zero, we want F
1-2
 = F
3
. If s is the length of each side of the
square, then:
(b) No. If q = Q/ , as found in part (a), then the net force on - q is not zero.
This is because F
1-2
 ? F
4
, as the following calculations show:
but