Exam I Practice Problems
1. Five charged particles are equally spaced around a semicircle of radius 100 mm,
with one particle at each end of the semicircle and the remaining three spaced
equally between the two ends. The semicircle lies in the region 𝑦 > 0 of an 𝑥𝑦plane, such that the complete circle is centered on the origin. Each particle
carries a charge of magnitude 6 nC. (Give all direction angles with respect to the
+𝑥-axis)
a. Draw and label a diagram that includes all electric field vectors at point 𝑂.
Include 𝑥- and 𝑦-components if necessary.
⃗ at point 𝑂.
b. Calculate the magnitude and direction of the net electric field 𝐄
c. If a 2 nC-charge is placed at point 𝑂, calculate the magnitude and
direction of the electric force 𝐅 that it experiences.
2. Three charges with magnitude 𝑞 are located at the following (𝑥, 𝑦) positions:
𝑞1 = +𝑞 at (0, −𝑎), 𝑞2 = −𝑞 at (0, +𝑎), and 𝑞3 = −𝑞 at (−𝑎, 0)
a. Draw and label a diagram that includes all electric field vectors at point 𝑃,
located at (+2𝑎, 0). Include 𝑥- and 𝑦-components if necessary.
⃗ at point 𝑃.
b. Calculate the magnitude and direction of the net electric field 𝐄
Your answer should include: 𝑘, 𝑞, and 𝑎.
c. If a +𝑄 charge is placed at point 𝑃, calculate the magnitude and direction
of the electric force 𝐅 that it experiences. Your answer should include: 𝑘, 𝑞,
𝑄, and 𝑎.
3. A proton moves at 4.50 × 105 m/s in the horizontal direction. It enters a uniform
vertical electric field with a magnitude of 9.60 × 103 N/C. Ignoring any
gravitational effects, find
a. the time interval required for the proton to travel 5.00 cm horizontally,
b. its vertical displacement during the time interval in which it travels 5.00 cm
horizontally, and
c. the speed after it has traveled 5.00 cm horizontally.
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4. Two source charges, 𝑞1 and 𝑞2 , are fixed along the corners of a triangle as
shown in the figure. The magnitude of 𝑞1 is 2 𝜇C, but its sign is not known. The
magnitude and sign of the charge 𝑞2 is unknown. Charge 𝑞3 is +4 nC, has a
mass 𝑚 = 4 × 10−14 kg, and experiences a net force 𝐹 entirely in the −𝑦direction.
a. Determine the signs of the charges 𝑞1 and 𝑞2 .
b. Calculate the magnitude of 𝑞2 .
c. Determine the magnitude of the acceleration that 𝑞3 experiences.
5. Three point charges are fixed at three corners of a rectangle. Charges 𝑞1 and 𝑞2
have unknown signs and magnitudes, and charge 𝑞3 = 3.2 𝜇C. A point charge
with a mass 𝑚 = 5.0 g and charge 𝑄 = 1.8 𝜇C is placed at the fourth corner and
⃗ = 700 m/s2 𝒊̂. Ignore any gravitational forces.
experiences an acceleration 𝐚
a. Determine the signs of 𝑞1 and 𝑞2 that will create this acceleration.
b. Draw a force diagram on charge 𝑄. Include components if necessary.
c. Use Newton’s Laws on charge 𝑄 to determine the magnitudes of 𝑞1 and
𝑞2 .
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6. Two glass half-rings are joined together with their centers coinciding with the
origin 𝑂 of the 𝑥𝑧-plane. The top ring has radius 3𝑅 and uniform charge density 𝜆,
and the bottom ring has radius 𝑅 and uniform charge density −6𝜆.
a. Determine the magnitude and direction of the net electric field at 𝑂 in
terms of 𝑘, 𝑅, and 𝜆.
b. Determine the magnitude and direction of the electric force on a +3𝑞 point
charge if it is placed at point 𝑂. Your answer should include 𝑘, 𝑅, 𝜆, and 𝑞.
7. A small thin plastic rod of length 𝐿 lies along the 𝑥-axis from 𝑥 = 0 to 𝑥 = 𝐿 and
has a charge distribution described by the expression
𝜆(𝑥) = 𝛼𝑥
where 𝑥 is measured in meters and 𝛼 > 0.
a. What are the SI units of 𝛼?
b. Determine the total charge 𝑄 on the rod.
c. Determine an expression for the magnitude and direction of the electric
field at point 𝑃, which is on the 𝑥-axis at a distance 𝑥 = 2𝐿 from the origin.
d. An infinite plane of surface charge +𝜎 is placed closely underneath the
rod. Determine an expression for the mass of the rod that will cause the
rod to levitate above the plane. Your answer should include: 𝛼, 𝐿, 𝜎, 𝜀0 ,
and 𝑔.
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where 𝛼 > 0 and 𝑟 is measured radially outward from the center of the sphere.
Concentric with this solid sphere is an insulating spherical shell with inner radius
𝑏 and outer radius 𝑐, and carries a uniform charge distribution with total charge
𝑄. Your answers may include: 𝑎, 𝑏, 𝑐, 𝑟,𝛼, 𝑄, or 𝜀0 .
a. What are the SI units of 𝛼?
b. Determine the total charge 𝑞 on the solid sphere of radius 𝑎.
c. Find the electric field (magnitude and direction) in the following regions:
i. 0 < 𝑟 < 𝑎
ii. 𝑎 < 𝑟 < 𝑏
iii. 𝑏 < 𝑟 < 𝑐
iv. 𝑟 > 𝑐
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distribution:
𝜌(𝑟) = 𝛼𝑟 2 , (0 < 𝑟 < 𝑎)
where 𝛼 is a constant greater than zero and 𝑟 is measured perpendicularly from
the axis joining the two cylinders. Concentric with this solid cylinder is a long
insulating solid cylinder with inner radius 𝑏 and outer radius 𝑐, and carries a
uniform charge distribution 𝜌0 . Your answers may include: 𝑎, 𝑏, 𝑐, 𝑟,𝛼, 𝜌0 , or 𝜀0 .
a. What are the units of 𝛼?
b. Find the electric field (magnitude and direction) in the following regions:
i. 0 < 𝑟 < 𝑎
ii. 𝑎 < 𝑟 < 𝑏
iii. 𝑏 < 𝑟 < 𝑐
iv. 𝑟 > 𝑐
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10. Two equipotential surfaces, 𝑆1 and 𝑆2 , exist in space, and are separated by a
distance of 2.0 cm. Point 𝑎 lies on 𝑆1 and points 𝑏 and 𝑐 lie along 𝑆2 . The electric
potential difference between the two surfaces is measured to be 8.0 kV, that is,
∆𝑉21 = 𝑉1 − 𝑉2 = +8 kV.
a. Which surface is at a higher potential, 𝑆1 or 𝑆2 ? Explain in 1-2 sentences.
b. Calculate the magnitude and direction of the uniform electric field between
the two surfaces.
c. If an external agent moves a charge 𝑞 = −4|𝑒| from point 𝑎 to 𝑐 at
constant speed:
i. Determine the work done by the external agent, 𝑊𝑒𝑥𝑡 .
ii. Determine the work done by the electric field, 𝑊𝑒 .
iii. Determine the change in potential energy, ∆𝑈𝑎𝑐 , of 𝑞 in units of
electron volts.
11. Two large, square metallic plates, each with a length of 50 cm, are placed 2 mm
apart and parallel to each other. A 3000 V potential difference is established
between each plate, giving each plate equal and opposite charge. Assume 𝐴 ≫
𝑑.
a. Calculate the surface charge density 𝜎 on each plate.
b. Calculate the charge 𝑞 on each plate.
c. Calculate the magnitude of the electric field 𝐸 between the plates.
d. What initial speed 𝑣𝑖 is required to fire an electron from the positive plate
towards the negative plate, to a distance . 95𝑑 where the electron
momentarily comes to rest?
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