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Capacitor with Oil

Task number: 2325

Plate capacitor with dimensions a = 10 cm, b = 15 cm and plate distance d = 0.2 mm touches the surface of oil with relative permitivity εr = 2.2 and density ρ = 760 kg m−3. We will connect the capacitor to DC voltage source with voltage U = 500 V. The oil then rises between the plates to height h.

Picture to task assignment

a) Determine the capacity of the capacitor.

b) Determine the electric field energy increase of the capacitor and the potential gravitational energy increase of the raised part of oil. The difference in height is denoted as dh.

c) Calculate h for given stable values.

d) Determine the value of voltage that would cause the oil to fill up the whole capacitor.

  • Analysis

    The capacity of the capacitor is directly proportional to the area of the plates and air permittivity and indirectly proportional to their distance before we connect them to the source.

    The oil level will rise to height h after we connect the source to the circuit. The capacity of the capacitor will increase since the permittivity of the environment with oil has increased. We can now imagine the capacitor as two parallel capacitors. One of them is fully filled with oil and the other is filled with air. The total capacity of parallel capacitors is obtained by adding up the capacities of each of the capacitors.

    Electric field energy of the capacitor is directly proportional to its capacity and the square of voltage. After we connect the capacitor to the source, its capacity increases and thus the energy of its electric field increases. We want to know how this energy changes according to capacity changes. We express the energy difference by derivative of energy with respect to capacity.

    Potential gravitational energy of oil in capacitor is directly proportional to its mass and the height of its centre of mass above the bottom edge of the capacitor. If the oil level rises then its potential gravitational energy rises as well. We can express this change of energy by the derivative of energy with respect to the height of oil surface.

    The source of voltage must work for the oil level to rise. The work done is equal to the potential gravitational energy increase and capacitor electric field energy. The work done is also directly proportional to the connected voltage and charge difference on capacitor. We can express the charge difference with capacitor capacity difference. If we have the work expressed using these two methods, we can calculate the height of the oil surface.

    If oil fills the whole capacitor, then the height of oil level is the same as the height of the capacitor. We can calculate the voltage from the equation that we formulated for oil surface height.

  • a) Hint: Capacity of Capacitor

    If the capacitor is partially filled with oil, we can imagine it as two parallel capacitors. One of them is filled with oil and the other is filled with air (more detailed solution of a similar situation can be found in Capaciter partially filled with dielectric).

    How do we calculate the capacity of parallel capacitors?

  • a) Solution: Capacity of Capacitor

    Before we connect the capacitor to a source, it had capacity

    C0=ε0abd.
    Capacitor separated into two parallel capacitors

    After we connect the capacitor to the source, the capacitor gets partially filled with oil. We can imagine it as two parallel capacitors (see picture). Their capacity is:

    Cvzduch=ε0a(bh)d Colej=ε0εrahd

    The total capacity of the capacitor can be calculated as the sum of capacities of each of the capacitors; the one filled with oil and the one filled with air.

    C=Cvzduch+Colej C=ε0a(bh)d+ε0εrahd

    We factor out common terms and adjust the equation.

    C=ε0ad(bh+εrh)

    The capacitor thus has capacity

    C=ε0ad(b+h(εr1))

    It is worth noticing how the capacity has changed from the original capacity C0 of the capacitor filled with air only. We will multiply the terms to get rid of the outer parentheses so that the first term is equal to the original capacity.

    C=ε0abd+ε0ah(εr1)d=C0+ε0ah(εr1)d

    The second term is linearly dependent on height h into which the oil rises. If oil rises by dh more, the capacity of the capacitor changes in this particular term. The increase in capacity denoted as dC will be:

    dC=ε0a(εr1)ddh.
  • b) Hint: Energy Change

    Capacitor electric field energy E changes with the change of the capacitor’s capacity and the potential gravitational energy Ep changes with rising surface of oil.

    Therefore we calculate the derivatives dEdC and dEpdh.

    The change of capacitor’s capacity with the change of height of oil surface has been derived in the previous section.

    Look into the note at the end of the task to see how we work with derivatives in this task.

  • b) Solution: Capacitor electric field energy increase

    This relation applies for the electric field energy inside the capacitor:

    E=12CU2.

    We want to know how this energy changes according to capacity. Therefore, we will calculate the derivative of energy with respect to capacity.

    dEdC=12U2

    The capacitor is connected to a stable voltage U which will therefore be considered to be a constant during derivation. We will express the energy difference.

    dE=12U2dC

    We will substitute for capacity difference dC from relation (1) which we derived in the solution of part a).

    dE=12U2ε0a(εr1)dhd

    If the surface of oil rises by dh, the energy of capacitor electric field changes by:

    dE=aε0(εr1)U22ddh.
  • b) Solution: Potential Gravitational Energy Difference

    We will calculate the potential gravitational energy using the following relation

    Ep=mgH.

    Attention! H is the height of the centre of mass of the object! The centre of mass is in height h2 in our case and so the following applies for potential energy:

    Ep=mgh2=Vϱgh2,
    Capacitor picture for oil volume calculation

    where V is the volume of the oil in the capacitor.

    We will calculate oil volume V = adh as the picture shows and we will substitute into relation (*).

    Ep=adhϱgh2=12adϱgh2

    We want to know how this potential gravitational energy changes according to the oil height difference. We will therefore calculate its derivative with respect to height.

    dEpdh=adϱgh

    The difference of potential gravitational energy equals:

    dEp=adϱghdh.
  • c) Hint: Oil Surface Height Calculation

    We can calculate the oil height from the following equation for work done by voltage source:

    dW=UdQ

    Think about how the work done affects capacitor energy.

  • c) Solution: Oil Level Height h Calculation

    We can calculate the oil surface height from work done by voltage source when it is raising the oil surface.

    Raise of oil surface by a small bit

    Source gives out constant voltage U and the oil surface is at height h. If we wanted the oil surface to rise a little bit more by dh, the source must do work dW.

     

    Since the voltage on the capacitor is constant, the work done is equal to

    dW=UdQ.

    The difference of charge with constant voltage can be expressed using capacitor capacity difference dQ = U dC.

    dW=U2dC.

    If the source does this work, the oil level rises by dh. This work will cause energy change. Since the oil level rose by dh, its potential gravitational energy rose by dEp. The capacitor’s capacity changed as well and thus the capacitor’s energy rose by dE.

    dW=dEp+dE

    Now we will compare both relations (2) and (3) for work calculation.

    U2dC=dEp+dE

    We have expressed all terms in the previous sections and now we will substitute them in.

    aε0(εr1)U2ddh=aε0(εr1)U22ddh+adϱghdh

    We will adjust the relation to express h. First we subtract the fractions

    aε0(εr1)U22ddh=adϱghdh.

    Then we will divide by a and dh

    ε0(εr1)U22d=dϱgh

    and we will express height h to which the oil rose in the capacitor.

    h=ε0(εr1)U22ϱgd2
  • d) Hint: Voltage Calculation

    Once oil fills the whole capacitor, what will the height of oil level be?

    Use the general result derived in the previous sectrion for calculation.

  • d) Solution: Voltage Calculation

    In section Oil Level Height Calculation, we have expressed height h to which the oil rises with the given voltage.

    h=ε0(εr1)U22ϱgd2

    We will use this result for voltage U1 calculation. The voltage is needed for the oil to fill the whole capacitor.

    If the oil fills the whole capacitor, its height h will be equal to the height of the capacitor b. Therefore, the following applies:

    Capacitor filled with oil
    h=b=ε0(εr1)U212ϱgd2.

    Now, we just need to express voltage U1.

    U21=2bϱgd2ε0(εr1) U1=2bϱgd2ε0(εr1)

    We have to connect the capacitor to voltage:

    U1=2bϱgε0(εr1)d.
  • List of Known Information and Numerical Calculation

    U = 500 V

    h = ? (m)
    a = 10 cm = 0.1 m U1 = ? (V)
    b = 15 cm = 0.15 m  
    d = 0.2 mm = 0.2·10−3 m

    From tables:

    εr = 2.2 ε0 = 8.85·10−12  C2N−1m−2
    ρ = 760 kg m−3 g = 9.8 m s−2

    h=ε0(εr1)U22ϱgd2=8.851012(2.21)500227609.8(0.2103)2m˙=4.46103m h˙=4,46mm U1=2bϱgε0(εr1)d=20.157609.88.851012(2.21)0.2103V U1˙=2901V˙=2.9kV
  • Answer

    The capacitor has capacity of C=ε0ad(b+h(εr1)).

    If the oil level rises by dh, the capacitor electric field energy changes by

    dE=aε0(εr1)U22ddh

    and the potential gravitational energy of the risen part of oil rises by

    dEp=adϱghdh.

    For the given values, the oil will rise to height

    h=ε0(εr1)U22ϱgd2˙=4.46mm.

    For the oil to cover the whole capacitor, we need to connect it to voltage of

    U=2bϱgε0(εr1)d˙=2.9kV.
  • Note: Derivatives and Fractions

    We often consider derivatives to be fractions in this task. This has to be mathematicaly proven but we will show only the „physical“ thought process behind it and why it is OK.

    We write, for example, the difference of potential gravitational energy at highschool as the following:

    ΔEp=mgΔh ΔEpΔh=mg

    If the increase Δh is very small (which means that it will be zero in its limit), then this applies:

    lim

    Therefore, if we work with small increments, we can „compare derivatives to quotients“

    \frac{\mathrm{\Delta} E_p }{\mathrm{\Delta} h\,}\dot=\, \frac{\mathrm{d} E_p }{\mathrm{d} h\,}

    and work with the derivatives like with fractions

    \frac{\mathrm{d} E_p }{\mathrm{d} h\,}=\, m g \, \mathrm{d} E_p\,=\, m g \, \mathrm{d} h
  • Link to Similar Tasks

    In task Dielectric lift you can find how the task would be solved had the capacitor been cylindrical.

    You can find the way how to calculate the capacity of a cylindrical capacitor with dielectric in task Capaciter partially filled with dielectric.

  • Other Ways of Oil Surface Height Calculation

    We could also calculate the oil surface height from the equation of electric and gravitational force like in task Dielectric lift

    .

    The result would be the same. We can see that from relation (**) from section Surface Height h Calculation.

    \frac{a \varepsilon_0 \left( \varepsilon_r -1\right)U^2}{2d}\mathrm{d}h\,=\, ad \varrho gh\,\mathrm{d}h\tag{**}

    After dividing by dh.

    \frac{a \varepsilon_0 \left( \varepsilon_r -1\right)U^2}{2d}\,=\, ad \varrho gh

    The gravitational force is on the right side of the equation. Since the mass of oil ism =  = adhρ, the gravitational force is equal to:

    F_G\,=\,mg\,=\,adh \varrho g

    The electric force is on the left side of the equation. The following has been derived in task Dielectric lift:

    F\,=\,\frac{1}{2}U^2\, \frac{\mathrm{d}C}{\mathrm{d}h}\,.

    In our case after substituting the capacity:

    F\,=\,\frac{a \varepsilon_0 \left( \varepsilon_r -1\right)U^2}{2d}\,.

    This applies in general:

    The components of force can be obtained by derivation of the potential energy with respect to the according coordinate.

    F_x\,=\, - \frac{\mathrm{d}E_p}{\mathrm{d}x}

    The force vector is therefore equal to

    \vec{F}\,=\, - \mathrm{grad}E_p\,.
Difficulty level: Level 3 – Advanced upper secondary level
Qualitative task
Tasks focused on analysis
Original source: Hubeňák, J. (1997). Řešené úlohy z elektřiny a magnetismu –
Proseminář z fyziky na střední škole a studující učitelství fyziky
v I. Ročníku. MAFY, Hradec Králové.
Zpracováno v diplomové práci Lenky Matějíčkové (2010).
×Original source: Hubeňák, J. (1997). Řešené úlohy z elektřiny a magnetismu – Proseminář z fyziky na střední škole a studující učitelství fyziky v I. Ročníku. MAFY, Hradec Králové.
Zpracováno v diplomové práci Lenky Matějíčkové (2010).
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