Ch 26.4 – Energy Stored in a Capacitor
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Ch 26.4 – Energy Stored in a Capacitor – charging a capacitor. Switch goes shut.  Battery establishes E-field in wires, charge builds up on the capacitor. Chemical energy in battery goes down and electrical potential energy stored in capacitor goes up.

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Switch goes shut.  Battery establishes E-field in wires, charge builds up on the capacitor.

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Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.4 – Energy Stored in a Capacitor – charging a capacitor

Switch goes shut.

 Battery establishes E-field in wires, charge builds up on the capacitor.

Chemical energy in battery goes down and electrical potential energy stored in capacitor goes up.

Electrical potential energy results from charge separation across capacitor’s plates


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.4 – Energy Stored in a Capacitor – charging a capacitor

To charge the capacitor, an external agent (battery) must do work to separate the charges.

Step by step, the battery “grabs” a small amount of charge dq off one capacitor plate and move it to the other.

At first, this requires no work, because the uncharged capacitor has no electric field to resist the movement of charge.

But, once dq has been transferred, the capacitor starts to develop a potential difference.

Now, to move more charge from one plate to the other, the battery must do some amount of work “dW,” to overcome the rising potential difference across the plates.

As more and more charge is transferred, the work required to transfer the same amount of charge, dq, goes up.


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.4 – Energy Stored in a Capacitor – charging a capacitor

Suppose q is the amount of charge on the capacitor at some instant during the charging process.

At the same instant, the potential difference across the capacitor is ΔV = q/C

The amount of work necessary to transfer dq amount of charge across this potential difference is

dW = ΔVdq = (q/C)dq


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.4 – Energy Stored in a Capacitor

Thus, the total amount of work required to charge the capacitor from q = 0 to a final charge of q = Q is

But, in an isolated system with no non-conservative forces, total mechanical energy must be conserved.

Therefore, the work done to charge the capacitor must equal the change in the system’s potential energy.


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.4 – Energy Stored in a Capacitor

Energy stored in a charged capacitor


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.4 – Energy Stored in a Capacitor

-It’s not obvious, but the potential energy stored in the capacitor actually resides in its electric field.

-This implies we should be able to solve the density of the energy stored in the field (J/m3).

-For a parallel plate capacitor, we already know:

-and, its capacitance is just:

-Substituting these into the purple equation,

-Dividing by the volume in between the plates of the capacitor (V=Ad), we get

Energy per unit volume in a capacitor (J/m3)


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.4 – Energy Stored in a Capacitor

-We don’t attempt it here, but it can be shown that this result is valid for any electric field!

Energy per unit volume in an electric field.

-In a very real sense, electric fields “carry” energy.


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

EG 26.4 – Rewiring two Charged Capacitors

Two capacitors, C1 and C2 (C1 > C2), are charged to the same initial potential difference, ΔVi. The charged capacitors are removed from the battery, and their plates are connected with opposite polarity, as shown. The switches, S1 and S2, are then closed.

  • Find the final potential difference ΔVf between a and b after the switches are closed.

  • Find the total energy stored in the capacitors before and after the switches are closed and determine the ratio of the final energy to the initial energy.


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.5 – Capacitors with dielectrics

A dielectric is something you stick in between the plates of a capacitor to change (increase) it’s capacitance.

The term comes from the fact that, at the atomic level, such materials can be polarized into arrays of dipoles.


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.5 – Capacitors with dielectrics

Common dielectric materials are: wax, paper, oil, polymers, fluid (electrolyte), etc.

Dielectrics are insulators.

Since you stick the dielectric material into the region of the capacitor’s E-field, it changes how “good” the capacitor is at establishing the E-field

 ΔV↓ for the same amount of Q on the capacitor plate.


What happens

What happens?


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.5 – Capacitors with dielectrics

Consider parallel-plate capacitor where ΔV0 = Q0/C0

Assume no battery is connected  Qcan’t change

When you stick a dielectric in between the plates 


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.5 – Capacitors with dielectrics

Consider parallel-plate capacitor where ΔV0 = Q0/C0

Assume no battery is connected  Qcan’t change

When you stick a dielectric in between the plates 

-where κ is a dimensionless constant called the “dielectric constant”


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.5 – Capacitors with dielectrics

-Q on the capacitor does not change

-Therefore:

-the capacitance is changed by a factor of κ.

-as κ goes up, C goes up.


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.5 – Capacitors with dielectrics

-For a parallel plate capacitor

  • To make capacitance ↑

    -decrease d

    -increase A

    -increase κ

- Only limited by “dielectric strength” of the dielectric


Example values of dielectric constant

Example values of dielectric constant

“Dielectric strength” is the

maximum field in the

dielectric before breakdown.

(a spark or flow of charge)


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

EG 26.5 – Energy stored before and after

A parallel-plate capacitor is charged with a battery to a charge of Q0. The battery is then removed, and a slab of material that has a dielectric constant κ is inserted between the plates. Identify the system as the capacitor and the dielectric.

Find the energy stored in the system before and after the dielectric is inserted.


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

EG 26.4 – Rewiring two Charged Capacitors

A parallel-plate capacitor is charged with a battery to a charge of Q0. The battery is then removed, and a slab of material that has a dielectric constant κ is inserted between the plates. Identify the system as the capacitor and the dielectric.

Find the energy stored in the system before and after the dielectric is inserted.

Before:

After:

Where did the energy go?


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

The combination of two equal charges of opposite sign, +q and –q, separated by a distance 2a

Every dipole can be characterized by it’s “dipole moment.”

- vector which points from –q to +q

-magnitude p = 2aq


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

What happens when we pop this baby in an external E-field?


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

What happens when we pop this baby in an external E-field?

-external field exerts F=qE on each charge

-net torque about the dipole’s center

-dipole rotates to “align” with the field


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

What happens when we pop this baby in an external E-field?

-external field exerts F=qE on each charge

-net torque about the dipole’s center

-dipole rotates to “align” with the field


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

but,

and

Thus:


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

but,

and

Thus:


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

  • The dipole and the external field are a system

  • -electric force is an internal conservative force

  • we can describe its work using a potential energy

    In other words, different configurations of the dipole-field system have different potential energies.


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

As the dipole aligns with the field, the system’s potential energy goes down.


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

-Work must be done to “un-align” the dipole from the field.

-in an isolated system, the work input must correspond to an increase in potential energy.


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

-Work must be done to “un-align” the dipole from the field.

-in an isolated system, the work input must correspond to an increase in potential energy.

W = ΔK + ΔU


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

-To rotate the dipole through some small angle dθ, an amount dW of work must be done.

but,


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

-To rotate the dipole through some small angle dθ, an amount dW of work must be done.

but,

-so, to rotate the dipole from θi to θf, the change in potential energy is:


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

Let’s define the zero potential energy as being when the dipole is at θ = 90,

when


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

Let’s define the zero potential energy as being when the dipole is at θ = 90,

when

We’ll use this reference energy as an anchor point.

At any time, we can write the system’s instantaneous potential energy, U, with respect to the zero-point potential energy.


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

Let’s define the zero potential energy as being when the dipole is at θ = 90,

when

We’ll use this reference energy as an anchor point.

At any time, we can write the system’s instantaneous potential energy, U, with respect to the zero-point potential energy.

But, we already know


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

Ch 26.6 – Electric Dipole in an Electric Field

Let’s define the zero potential energy as being when the dipole is at θ = 90,

when

We’ll use this reference energy as an anchor point.

At any time, we can write the system’s instantaneous potential energy, U, with respect to the zero-point potential energy.

But, we already know


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

EG 26.6 – The Water Molecule

A water molecule has an electric dipole moment of 6.3x10-30 Cm. A sample contains 1021 water molecules. All of the dipoles are oriented in the direction of an external E-field, which has a magnitude of 2.5x105 N/C.

How much work is required to rotate all the dipoles from this orientation (θ = 0) to one in all the dipoles are perpendicular to the external field (θ = 90)?


Switch goes shut battery establishes e field in wires charge builds up on the capacitor

EG 26.6 – The Water Molecule

A water molecule has an electric dipole moment of 6.3x10-30 Cm. A sample contains 1021 water molecules. All of the dipoles are oriented in the direction of an external E-field, which has a magnitude of 2.5x105 N/C.

How much work is required to rotate all the dipoles from this orientation (θ = 0) to one in all the dipoles are perpendicular to the external field (θ = 90)?


Example p26 9

Example P26.9

When a potential difference of 150 V is applied to the plates of a parallel-plate capacitor, the plates carry a surface charge density of 30.0 nC/cm2. What is the spacing between the plates?


Example p26 21

Example P26.21

  • Four capacitors are connected as shown in Figure P26.21.

  • Find the equivalent capacitance between points a and b.

  • Calculate the charge on each capacitor if ΔVab = 15.0 V.


Example p26 27

Example P26.27

Find the equivalent capacitance between points a and b for the group of capacitors connected as shown in Figure P26.27. Take C1 = 5.00 μF, C2 = 10.0 μF, and C3 = 2.00 μF.


Example p26 35

Example P26.35

A parallel-plate capacitor is charged and then disconnected from a battery. By what fraction does the stored energy change (increase or decrease) when the plate separation is doubled?

. Therefore, the

,


Example p26 43

Example P26.43

Determine (a) the capacitance and (b) the maximum potential difference that can be applied to a Teflon-filled parallel-plate capacitor having a plate area of 1.75 cm2 and plate separation of 0.040 0 mm.


Example p26 59

Example P26.59

A parallel-plate capacitor is constructed using a dielectric material whose dielectric constant is 3.00 and whose dielectric strength is 2.00 × 108 V/m. The desired capacitance is 0.250 μF, and the capacitor must withstand a maximum potential difference of 4 000 V. Find the minimum area of the capacitor plates.


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