AP Free Response Question
2011 B5
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In the experimental setup represented below, a very small plastic sphere of mass
m
with charge
q
is allowed to fall under the influence of gravity between two parallel metal plates separated by a fixed distance
L
. A variable potential difference may be applied between the two plates. The experiment is conducted inside a vacuum chamber.
(a) A potential difference of magnitude
V
is applied between the top and bottom plates such that the sphere falls at constant speed
v
. Derive an expression for the magnitude of the charge
q
on the sphere. Express your answer in terms of
m
,
L
,
V
, and fundamental constants, as appropriate.
The experiment is performed many times with spheres of identical known mass but different unknown charges, each time adjusting the potential difference
V
to the value needed so that the sphere falls at constant speed
v
. The magnitudes of the charges are calculated from the measured values of the potential difference. The data is plotted below as a function of the magnitude of
V
.
(b) Provide a physical explanation for the gap observed in the data between potential differences of 1700 V and 2800 V.
(c) If the value of L is 0.050 m, calculate the mass
m
of the spheres.
A uniform magnetic field of magnitude
B
, directed into the page, is now applied in the bottom half of the region between the plates, as shown in the figure below. The experiment is repeated, with the potential difference adjusted again so that the charged sphere falls with constant speed prior to entering the magnetic field.
(d) i. Describe the motion of the sphere as it travels through the magnetic field.
(d) ii. Describe how the motion could be used to determine the sign of the charge.
(e) Derive an expression for the minimum value of
B
needed to prevent the sphere from reaching the bottom plate. Express your answer in terms of
m
,
q
,
v
,
L
, and fundamental constants, as appropriate.
Topic Formulas
Description
Published Formula
capacitance
Coulomb's Law
electric field
electric potential energy
energy stored in a capacitor
Faraday's Law
friction
gravitational potential energy
Hooke's Law
magnetic field around a current-carrying wire
magnetic flux
magnetic force on a current-carrying wire
magnetic force on a moving charge
Newton's 2nd Law
Newton's Law of Universal Gravitation
parallel-plate capacitor
potential and electric field strength
potential due to a collection of point charges
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