1 Electric homogeneous field 1

A capacitor for the voltage range of \(100 \ kV\) is to be realised with a dielectric in which a maximum field strength of \(E = 30 \ \frac {kV}{cm}\) is permissible. Calculate the capacitance of the capacitor.

Task section 1:

The dielectric is first used in an (ideal) plate capacitor.

a1)

Sketch the arrangement.

b1)

Sketch the field lines in the arrangement.

c1)

Where is the location of the maximum field strength?

d1)

Sketch the course of the electric field strength from one capacitor plate to the other \(\left | E(x) \right |\), parallel to the capacitor plates \(\left | E_{Diel}(z) \right |\) and in the capacitor plates \(\left | E_{Pl}(z) \right |\).

e1)

Draw the 50%- equipotential line.

f1)

Calculate the minimum plate spacing \(d_{Pl,min}\) for the above boundary conditions.

Task section 2:

Now the dielectric between coaxial cylinders is to be inserted.
\(\left (E_r(r)=\frac {U}{r\cdot \ln \left (\frac {a}{b}\right )};\ r_a: Outer radius; r_i: Radius \ of the \ inner cylinder\right )\)

a2)

Sketch the arrangement.

b2)

Sketch the field lines.

c2)

Where is the location of the maximum field strength?

d2)

Sketch the course of the electric field strength \(E_r(r)\) from \(r=r_i\) to \(r=r_a\).

e2)

Draw the 50

f2)

Calculate the minimum outer radius \(r_a\) for \(r_i = 1;\ 2;\ 3;\ 4;\ 5;\ 10 \ cm\) and the corresponding thickness of the dielectric \(d_i\).

1.1 Lösung:

Hier entsteht eine Musterlösung...

2 Electric homogeneous field 2

Between the plates of the adjacent capacitor are two insulating materials with dielectric numbers \(\varepsilon _{\mathrm {r1}}\) and \(\varepsilon _{\mathrm {r2}}\). The capacitor is connected to a battery via switch \(S\).

The following dimensions are given: Plate sub-areas \(A_1 = A_2 = 100 \ cm^2; U_{\mathrm {B}}=100 \ V;d = 1 \ cm\).

a)

The switch is closed. Let \(\varepsilon _{\mathrm {r1}}=\varepsilon _{\mathrm {r2}}=1\). How large are the capacitor charge \(Q\), the electric flux density \(D\), the electric field strength \(E\) and the capacitance \(C_{\mathrm {AB}}\)?

b)

The switch is closed. Let \(\varepsilon _{\mathrm {r1}}=1, \varepsilon _{\mathrm {r2}}=3\). How has the total capacitance changed in general compared to point a)? The values \(Q_1\) and \(Q_2\), \(D_1\) and \(D_2\), and \(E_1\) and \(E_2\) must also be determined. What voltage \(U_{AB}\) is established at the capacitor?

c)

In the arrangement according to point a), another dielectric is introduced after the switch \(S\) is opened, so that the following applies: \(\varepsilon _{\mathrm {r1}}=1, \varepsilon _{\mathrm {r2}}=3\). What voltage \(U_{\mathrm {AB}}\) is established at the capacitor?

2.1 Lösung:

Hier entsteht eine Musterlösung...

3 Electric homogeneous field 3

A capacitor for the voltage range of 100 kV is to be constructed using a dielectric in which a maximum field strength of E = 30 kV/cm is permitted.
The dielectric is first used in an (ideal) Plattenkondensator.

a1)

Sketch the arrangement.

b1)

Sketch the field lines in the arrangement.

c1)

Where is the location of maximum field strength?

d1)

Sketch the course of the electric field strength from one capacitor plate to the other \(\left | E(x) \right |\), parallel to the capacitor plates \(\left | E_{\mathrm {Diel}}(z) \right |\) and in the capacitor plate \(\left | E_{\mathrm {Pl}}(z) \right |\).

e1)

Draw the 50%- equipotential line.

f1)

Calculate the minimum plate spacing \(d_{\mathrm {Pl,min}}\) for the above boundary conditions.

Now the dielectric between coaxial cylinders should be inserted.

\(\left (E_{\mathrm {r}}(r)=\frac {U}{r\cdot \ln \left (\frac {a}{b}\right )};\ r_{\mathrm {a}}: Outer radius; r_{\mathrm {i}}: Radius \ of the \ inner cylinder\right )\)

a2)

Sketch the arrangement.

b2)

Sketch the field lines.

c2)

Where is the location of maximum field strength?

d2)

Sketch the course of the electric field strength \(E_{\mathrm {r}}(r)\) from \(r=r_{\mathrm {i}}\) to \(r=r_{\mathrm {a}}\).

e2)

Draw the 50%- equipotential line.

f2)

Calculate the minimum outer radius \(r_{\mathrm {a}}\) for \(r_{\mathrm {i}} = 1;\ 2;\ 3;\ 4;\ 5;\ 10 \ cm\) and the corresponding thickness of the dielectric. \(d_{\mathrm {i}}\).

3.1 Lösung:

Hier entsteht eine Musterlösung...

4 Charge carrier velocity

A current \(I = 8 \ A\) flows through a copper wire with a cross-sectional area \(A = 1 \ mm^2\) and a length \(L = 10 \ m\). One \(mm^3\) of copper contains \(8.5\cdot 10^{19}\) atoms. It can be assumed that 1 electron per atom is involved in charge transport. \((\vartheta = 20 \ ^\circ C)\).

a)

Determine the drift velocity of the electrons in the copper wire.

b)

What is the electric field strength \(E\) in the copper wire?

c)

How high is the voltage drop \(U\) in this copper wire?

d)

What is the resistance of the copper wire under the specified boundary conditions? What resistance does the wire assume when heated to \(\vartheta _{\mathrm {w}} = 180 \ ^\circ C\)? How much is the percentage increase in resistance?

4.1 Lösung:

Hier entsteht eine Musterlösung...