A parallel-plate capacitor has plates of area 0.12 m^2 and a separation of 1.2 cm . A battery ch

A parallel-plate capacitor has plates of area 0.12 m^2 and a...

Key Concepts

Parallel-Plate Capacitor

A parallel-plate capacitor is an arrangement of two conductive plates separated by a gap in which an electric field can be established. Its basic properties include the area of the plates and the distance between them, which determine its capacitance, and it serves as a fundamental model in electrostatics and circuit theory for storing electric energy.

Dielectric Material

A dielectric is an insulating material that, when placed within a capacitor, reduces the effective electric field between the plates and increases the capacitance. Its effectiveness is characterized by its dielectric constant, a dimensionless factor that quantifies how much the material increases the capacitor's ability to store charge compared to a vacuum.

Capacitance and Its Calculation

Capacitance is a measure of a capacitor's ability to store an electric charge per unit potential difference. For a basic parallel-plate capacitor, the capacitance is usually calculated using the formula C = (??A)/d, where ?? is the vacuum permittivity, A is the area of one of the plates, and d is the separation between the plates. In the presence of a dielectric, the capacitance is modified based on the dielectric constant and the geometry of the configuration.

Charge Conservation in Capacitors

When a capacitor is charged to a certain voltage and then disconnected from its battery, the free charge on the plates remains constant. This principle of charge conservation plays a crucial role when modifications, such as the insertion of a dielectric, are made to the capacitor, as it affects how the potential difference and stored energy adjust to the new conditions.

Electric Field Distribution

The electric field in a capacitor is defined as the force per unit charge between the plates. The introduction of a dielectric alters this distribution by reducing the effective field within the material compared to the field in the air or vacuum regions. Understanding this variation is key to analyzing how the overall behavior of the capacitor changes with the presence of different materials.

Potential Difference Change

The potential difference, or voltage, across the plates of a capacitor is determined by the amount of charge present and the capacitance of the system. When a dielectric is inserted into a disconnected capacitor, the constant charge and altered capacitance lead to a change in the voltage between the plates, a concept that underscores the interplay between electrical energy storage and material properties.

Work Done in Inserting Dielectric

Inserting a dielectric into a capacitor involves external work, which may result in a change in the energy stored in the capacitor. This work is necessary to overcome the forces acting on the dielectric due to the non-uniform electric field, and it is a critical concept in energy analysis and the study of electrostatic forces in capacitive systems.

Transcript

00:01 Here we're going to look at a capacitor plate arrangement that initially has no dielectric in it.

00:08 And then we are going to disconnect the battery, which is initially set at 120 volts.

00:15 And we are going to place a dielectric between the two plates separated with equal gaps on either side, each of 4 millimeters.

00:27 So this is a nice symmetric arrangement.

00:33 And we're going to ask, how does this change the capacitance? so there are a couple things to realize is that there are two ways that you can actually set up this dielectric.

00:46 One of which is to leave the battery connected.

00:49 And if you put the slab in between, more charge will dance onto the plates due to the battery.

00:58 Charge inside the dielectric.

01:01 So let's make the positive plate above and the negative plate below.

01:08 What will happen is if you've disconnected the battery, the charge remains the same.

01:16 There's no place for it to go.

01:17 It's going to stick on those plates, and that's the second way you can set up this arrangement.

01:24 But in either case, the capacitance is going to change because of the bound charge that builds up on that dielectric.

01:36 So it actually has an electric field inside of it that propels charges to go to the surface.

01:46 So that's called bound charge.

01:49 And for a uniform electric field that occurs in the plates, that bound charge, occurs at the surface.

01:59 There are bound charges inside, but it cancels.

02:03 They cancel each other.

02:05 So what we're going to have to find is the new capacitance of this arrangement, and we're going to have to find the potential energy, new potential energy stored.

02:16 So we're going to start with our initial condition and look at some capacitor relationships.

02:23 We know that the capacitance is, to the area times the permittivity of free space over the separation.

02:34 So that is a geometric calculation that we can do.

02:41 And we want to use the permittivity of free space in there, a well -known constant.

02:50 And the spacing is 12 millimeters or 1 .2 centimeters, but we want that in meters.

02:57 And that works out to be in the pico -faraday range.

03:06 I will write the pico -faraday out as 88 .5 times 10 to the minus 12.

03:14 In terms of the potential and the electric field, though, we know that capacitance is defined to be the ratio between the charge on the plates divided by potential difference.

03:31 So c equals q over delta v, we can now find the q that's on the plates.

03:41 And this is a fixed quantity.

03:45 I will point that out as we go along.

04:02 And we won't forget our pico, and that turns out to be in the nano -culum range.

04:09 I will write out the nano as 10 to the minus 9 quillums.

04:13 But the thing to realize is that does not change between the inserting of the slab and the initial condition.

04:23 And the reason for that is the battery is not hooked up.

04:29 If it were left hooked up, then you could have the capacitance change by having the charge change.

04:37 So what's going to change is the potential difference across the plates.

04:46 And let me just remind us of that.

04:49 Q remains the same, and it's the delta v will change to make the capacitance bigger.

05:06 And essentially the capacitance is bigger because of those stored bound charges in your dielectric slab.

05:16 Okay, now to work out the new capacitance, so this is all initial stuff.

05:24 To work out the final situation, i like to use something called the d field.

05:33 The d field stands for displacement.

05:36 And how it differs from the electric field is it depends just on what's called the free charge.

05:45 So there's a distinction between free charge that is not bound inside and touch.

05:51 Insulator and in this case on the conducting plates, meaning that it's free to move around.

06:00 It doesn't have to be on a conductor.

06:02 It could be floating around as a point charge in space, but usually with parallel plates, that free charge is sitting on your conductor plates.

06:11 And the displacement field depends only on that.

06:16 And it is related to the electric field through the dielectric medium, the dielectric constant times epsilon 0 times e is equal to d.

06:33 So remember that the free charge is fixed, and we can find the d field fairly readily through equation 1, which is d equals q over a.

06:48 And then we can find the electric field in our three regions.

06:52 The gap, we'll call that g for gap, and d for die -electric...

Please give Ace some feedback