1 Material dependence of resistance R
A copper wire has the following properties:
- Length: \( l = 5\,mathrm{m} \)
- Cross-sectional area: \( A = \SI {1}{\milli \meter \squared } \)
- Specific resistance of copper: \( \rho = \SI {1.68e-8}{\ohm \meter } \)
Calculate \( R \) of the wire.
2 Electrical resistance
A spherical hollow structure consists of two ideally conductive shells – an inner one with radius \( a \) and an outer one with radius \( b \).
The space between them is completely filled with a homogeneous, conductive material that has electrical conductivity \( \kappa \) (in \(\mathrm {S}\,\mathrm {m}^{-1}\)).
A voltage \[ U = \varphi _1 - \varphi _2 \] is applied between the conductive shells by a voltage source.
The current can spread unhindered due to the complete filling and ideal contacts.
Calculate the electrical resistance \( R \) between the two spherical shells as a function of \( a \), \( b \) and \( \kappa \).
3 Electrical resistance of a coated wire
A round aluminium wire has a radius of \( a = \SI {0.5}{\milli \meter } \). It is coated with a thin layer of gold. \( b = \SI {0.1}{\milli \meter } \) Thickness coated.
The following values apply to the specific electrical conductivity and temperature coefficient of the two metals:
- Aluminium: \[ \kappa _\text {Alu} = \SI {35}{\meter \per (\ohm \milli \meter \squared )}, \alpha _\text {Alu} = 3{,}5 \cdot 10^{-3}\,\si {\per \kelvin } \]
- Gold: \[ \kappa _\text {Gold} = \SI {44}{\meter \per (\ohm \milli \meter \squared )}, \alpha _\text {Gold} = 3{,}6 \cdot 10^{-3}\,\si {\per \kelvin } \]
Given: Wire Length: \( l = \SI {1.5}{\meter } \) Temperature: \( T = \SI {25}{\celsius } \) (Initial temperature)
- Calculate the electrical DC resistance of the aluminium core and gold coating at \( T = \SI {25}{\celsius } \).
- How does the respective direct current resistance change when the temperature rises to \( T = 90 \celsius \)?
4 Real power sources
A real current source has an open-circuit voltage \( U_0 = 9\,\si {\volt } \) and an internal resistance \( R_\text {i} = 1\,\si {\ohm } \).
a) How large is the terminal voltage when a load resistance of \( R_l = 4\, \si {\ohm } \) is connected? b) How large is the current through the load?
5 Voltage source
Given is a network consisting of a voltage source with \( U = \SI {10}{\volt } \), a series resistor \( R_1 = \SI {2}{\ohm } \) and a parallel resistor \( R_2 = \SI {4}{\ohm } \).
Calculate currents and voltages in the network.
6 Capacitance and capacitor
Part 1: Single capacitor
Given: \[ C = \SI {10}{\pico \farad }, U = \SI {5}{\volt } \]
- Calculate the load \( Q \)
- Calculate the energy stored in the capacitor.
Part 2: Two capacitors in series
Two capacitors \( C_1 \) and \( C_2 \) are connected in series.
\[ C_1 = \SI {4}{\micro \farad }, C_2 = \SI {6}{\micro \farad }, U_\text {ges} = \SI {12}{\volt } \]
- Calculate the total capacity \( C_\text {ges} \)
- How great is the excitement about \( C_1 \)?
7 Inductance and coil
Part 1: Energy in the coil
Given is a coil with the inductance \[ L = \SI {2}{\henry } \] and a current \[ I = \SI {3}{\ampere }. \]
- Calculate the energy stored in the magnetic field of the coil.
- How does the energy change when the current is reduced to \( I = \SI {0.5}{\ampere } \)?
Part 2: Inductance of a cylindrical coil
A cylindrical coil has \[ N = 500 \text { Windings}, l = \SI {20}{\centi \meter }, A = \SI {5}{\centi \meter \squared } \] The inside of the coil is filled with air.
Calculate the inductance \( L \) of the coil.
8 Inductance, magnetic flux density and magnetic field energy
An air-filled, long cylindrical coil conductor has the following properties:
- Number of turns: \( N = 500 \)
- Length of the coil: \( l = \SI {0.5}{\meter } \)
- Cross-sectional area: \( A = 4 \cdot 10^{-4} \, \si {\meter \squared } \)
- Current: \( I = \SI {2.5}{\ampere } \)
Given: Magnetic field constant: \( \mu _0 = 4\pi \cdot 10^{-7} \, \si {\henry \per \meter } \)
- Calculate the magnetic flux density \( \vec {B} \) inside the coil.
- Determine the inductance \( L \) of the coil.
- Calculate the energy \( E_m \) stored in the magnetic field.
