PHY2049 Fall 2005
Prof. Andrew Rinzler
Prof. Paul Avery
Oct. 19, 2005
PHY2049, Fall 2005
Exam 2 Solutions
1. A wire made of a particular material has resistance R. Another wire made of the same material
that has 1/4 the length and 1/4 the diameter has resistance?
(1) 4R
(2) 16R
(3) R / 2
(4) R / 4
(5) R / 16
Resistance is proportional to length and inversely proportional to cross-sectional area. Since the
length is 1/4 of the old length and the area is 1/16 the old area, the new resistance is 4 times as
great or 4R.
2. A 2.00 kW heating element from a hair drier designed to operate at 120 V has a length of 80.0
cm. A 10.0 cm section of the element is removed. What power in watts is used by the now shortened heating element at 120 V?
(1) 2290
(2) 1750
(3) 2250
(4) 1810
(5) need the resistivity
The new resistance Rnew is 7/8 the old resistance R. The total power, which is V 2 / Rnew , is thus
8/7 times the old power or 2290 W.
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PHY2049 Fall 2005
3. For the circuit shown the resistors all have resistance R. The battery has internal resistance r.
What is the power dissipated by the battery?
(1) 2V 2 / ( R + 2r )
(2) 4V 2 / ( 3R + 4r )
(3) 2V 2 / ( R + 4r )
(4) 4V ( 3R + 4r )
(5) 2V ( R + 2r )
The total external resistance can be found from the rules of adding parallel and series resistances to be R / 2. The total resistance is then r + R / 2. The total power can be calculated from
V 2 / ( r + R / 2 ) = 2V 2 / ( R + 2r ) .
!
4. A current carrying loop (Area vector A , normal to the loop by right hand rule) forms part of a
simple electric motor and rotates about an axis that bisects the loop in a uniform magnetic field.
In order to maximize the average torque generated by the motor the commutator should flip the
current direction in the loop when
axis
! !
(1) A " B
! !
(2) A ⊥ B
!
(3) A is 45o from
!
(4) A is 45o from
(5) never
!
B
!
!
B approaching B
!
!
B past B
free to
rotate
i
We want to flip the commutator when A is parallel to B because that ensures that the torque is
always in the direction of the rotation and is thus at its maximum average value.
5. Three wires lie along the x axis (perpendicular to the page) as shown. Wires A and B each
carry current i into the page. When wire C carries no current, wire B experiences a force F. With
no other currents changed, in order for wire B to experience a force 3F in the same direction as
the original F, the current in wire C must be made?
(1) 2i, out
(2) 3i, out
(3) 3i, in
(4) i, in
(5) i / 4, out
To increase the force, the current in C must be out of the page, by the right hand rule. To add
2F, the current must be twice as large as A since the distance is the same.
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PHY2049 Fall 2005
6. Two long ideal solenoids (with radii 20mm and 30 mm, respectively) have the same number
of turns of wire per unit length. The smaller solenoid is mounted inside the larger, along a common axis. The magnetic field within the inner solenoid is zero. If the current in the outer solenoid
is I, what is the current in the inner solenoid?
(1) I
(2) 2I / 3
(3) I / 3
(4) 2I
(5) I / 2
The B field is the same regardless of the diameter, so the current is the same.
7. A solid circular cylinder of radius R carries a uniformly distributed current I. A wire runs parallel to the cylinder along the surface (see figure). Find the magnitude and direction of the current in the wire such that the resultant magnetic field at the point marked with a cross (located at
r = R / 2) is zero.
Wire
(1) I / 4, same
(2) I / 2, opposite
(3) I, same
(4) 2I, opposite
(5) 4I, same
At r = R / 2, we find the B field from the cylinder from Ampere’s law, where the enclosed current
is clearly 1/4 the total current. Since the point is equidistant from the center of the cylinder and
the wire, the current in the wire must be the same, or I / 4. Simple application of the right hand
rule shows that it must be in the same direction.
8. In the circuit shown, the battery has potential 20V, the switch S is open and the capacitor
( C = 2.5µ F ) is initially uncharged. The switch is then closed long enough to allow the capacitor
to fully charge. If the switch is now opened, how much time (in µ sec ) is required for the potential difference across the capacitor to reach 2 V?
S
(1) 23.0
(2) 1.05
(3) 5.2
(4) 15.1
(5) 27.6
6Ω
6Ω
C
6Ω
The parallel resistor combination is 4Ω, so the RC time is 4 × 2.5 = 10µs. The discharge equation is 2 = 20e −t /10 , so t = 10ln2 = 23µs.
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PHY2049 Fall 2005
9. The three light bulbs in the picture are identical. If the brightness of bulb A is PA, what is the
brightness of bulb B?
(1) PA / 4
(2) PA / 2
(3) PA
(4) 2PA
(5) 4PA
A
B
C
The current in the lower branch is 1/2 the current in the first branch, so the power is 1/4 as
large.
10. A proton moves in a direction having an angle of 30o relative to a uniform magnetic field of
3.0 T with an energy of 10 MeV ( 1eV = 1.6 ×10−19 J). What is the radius (in meters) of the helix
describing its path?
(1) 0.076
(2) 0.15
(3) 0.30
(4) 0.13
(5) 0.18
Only the perpendicular component of velocity counts when calculating radius, so the radius is
R = mv⊥ / eB , where v⊥ = v sin 30° . We can find v from v = 2 K / m , where K is the kinetic energy in joules. This procedure yields R = 0.076 m.
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