Physics 2 (9 CFU)
Name
Surname
Exercise n.1
A.A. 2011-2012
n. matricola
10.2.2011
An electrical charge q is distributed over an insulating cylinder having radius R=1 m and height L>>R. The
electrical charge density varies linearly with the distance from the cylinder axis: it is zero on the axis and
reaches the value of 0 = 10-6 C/m3 at the cylinder surface. Calculate the electrostatic field at a distance
r=10 cm from the cylinder axis.
Charge density varies according the law:
 r    0
r
,
R
We use as a Gaussian surface a cylinder of radius r, therefore
r
r
0
0
q'    2rldr    0
20 r 3l
r
2rldr 
R
3R
Per il teorema di Gauss, all’interno della sfera il campo elettrostatico sarà:
 E  
20 r 3l

0
3R 0
q'
0r 2
 E
 376V
3R 0
Exercise n. 2
A conducting wire, 1 m long, consists of a material having a resistivity ρ=10-6Ω×m. The wire has a circular section of
radius 1mm except for a short portion, 1 cm long, where the radius is reduced to 0.1mm. The two ends of the wire are
connected to a variable e.m.f. supply. Supposing that the maximum power dissipation on the thin section of the wire,
before melting, is 10 W, calculate the maximum allowed e.m.f.
The current flowing through the wire is:
𝜌𝐿
𝜌𝑙
V  I  Rtot , where 𝑅𝑡𝑜𝑡 = 𝑅𝑡ℎ𝑖𝑐𝑘 + 𝑅𝑡ℎ𝑖𝑛 = 𝜋𝑅
2 + 𝜋𝑅 2 ≅ 0.31Ω + 0.31Ω = 0.62 Ω
The power dissipated on the thin section is.
Δ𝑉 2
𝑃 = 𝐼 2 𝑅𝑡ℎ𝑖𝑛 = (
) 𝑅𝑡ℎ𝑖𝑛
𝑅𝑡𝑜𝑡
From this we finally achieve:
𝑃
Δ𝑉 = 𝑅𝑡𝑜𝑡 √
≅ 3.5𝑉
𝑅𝑡ℎ𝑖𝑛
Exercise n. 3
A solenoid having indefinite lenght and radius r=10 cm, with N, number of turns per unitary lenght equal to
103 turns/m is powered by a current supply. The current through the solenoid varies from 0 A to 10 A in 10 s
with a quadratic dependence on time. A single turn coil having radius 20 cm and an electrical resistance 10
Ω is placed around the solenoid. The axis of the coil is parallel to the axis of the solenoid. The distance
between the two axes is 2 cm. Calculate the current flowing through the coil 5 s after the current supply is
switched on...
For the geometrical conditions described in the text the solenoid is fully inside the coil. The internal magnetic
field varies according the law:
B  N0i  N0 Kt 2 where K=0.1 A/s2

2r 2 N 0 Kt
d
1 d
iR    B  i  
r 2 N 0 Kt 2  
 4 A
dt
R dt
R



The current direction through the coil is opposite to the current direction in the solenoid...
Exercise n. 4
Let’s suppose to have a e.m.f. supply of 12 V and three identical lamps, each having a resistance of 3 Ω. In
which configuration of series and parallel should the three lamps be connected in order to achieve the
maximum luminosity? In such a configuration which will be the number of photons emitted by the unit of
time? Suppose that the lamps emit monochromatic radiation having wavelength λ=0.5 µm.
The three lamps have to be connected in parallel to have the maximum power dissipated.
𝑃=
𝑉2
𝑅
= 144 𝑊
The energy associated with the emitted photons is:
𝑃 = 𝑁ℎ𝜈
Therefore:
𝑃𝜆
𝑁=
≅ 3.6 × 1020
ℎ𝑐