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Section 19.1 in Matter and Interactions (4th edition) | Section 19.1 in Matter and Interactions (4th edition) | ||
- | [[184_notes: | + | / |
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===== Capacitors in a Circuit ===== | ===== Capacitors in a Circuit ===== | ||
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==== Definition of Capacitance ==== | ==== Definition of Capacitance ==== | ||
- | **Capacitance** is defined as the proportionality constant of the potential difference between the plates of the capacitor (|ΔV|) and the amount of charge that is on one plate (|Q|) - if the charge increases, the potential difference should increase and vice versa. Note that since one plate will always have a charge of +Q and one will will have a charge of -Q (because of conservation of charge), the |Q| will be the same no matter which plate you are looking at. In terms of an equation, this means that: | + | **Capacitance** is defined as the proportionality constant of the potential difference between the plates of the capacitor (|ΔV|) and the amount of charge that is on one plate (|Q|) - if the charge increases, the potential difference should increase and vice versa. Note that since one plate will always have a charge of +Q and one will have a charge of -Q (because of conservation of charge), the |Q| will be the same no matter which plate you are looking at. In terms of an equation, this means that: |
|Q|=C|ΔV| | |Q|=C|ΔV| | ||
- | The units of capacitance are then coulombs/ | + | The units of capacitance are then **coulombs/ |
==== Finding Capacitance ==== | ==== Finding Capacitance ==== | ||
We can actually find an expression for capacitance specifically for parallel plates using the [[184_notes: | We can actually find an expression for capacitance specifically for parallel plates using the [[184_notes: | ||
- | {{ 184_notes:5b_diagram_solution.jpg?300}} | + | [{{ :184_notes:e-field_between_parallel_plates_new.png?300|Electric field between two parallel plates. The dashed blue arrows represent the E-field from the negatively charged plate, and the solid red arrows represent the positive plate. |
- | === Deriving the Capacitance for Parallel Plates === | + | ==== Deriving the Capacitance for Parallel Plates |
We start by considering the electric field between the plates. We know that for two parallel plates, there is an electric field in the middle that points directly from the positive plate to the negative plate (except near the edges where the field bends slightly out). Outside of the plates, the electric field is zero because the contributions from the negative and positive plates will cancel. This means that there is a very strong electric field in the middle of the plates and a very small electric field outside the plates. If we __//assume that the area of the plates is very large compared to separation between them//__, then the electric field between the plates is approximately constant. It turns out that the magnitude of this electric field is related to only the charge density (charge per area) on the plates // | We start by considering the electric field between the plates. We know that for two parallel plates, there is an electric field in the middle that points directly from the positive plate to the negative plate (except near the edges where the field bends slightly out). Outside of the plates, the electric field is zero because the contributions from the negative and positive plates will cancel. This means that there is a very strong electric field in the middle of the plates and a very small electric field outside the plates. If we __//assume that the area of the plates is very large compared to separation between them//__, then the electric field between the plates is approximately constant. It turns out that the magnitude of this electric field is related to only the charge density (charge per area) on the plates // | ||
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If we want the change in potential from the bottom to the top plate, then we would want to integrate along the vertical direction, which means we would want to change: | If we want the change in potential from the bottom to the top plate, then we would want to integrate along the vertical direction, which means we would want to change: | ||
→dr=dyˆy | →dr=dyˆy | ||
- | and our bounds would be from 0 to d. Plugging in what we found for →E | + | and our bounds would be from 0 to d. Plugging in the electric field between the plates |
ΔV=−∫d0−σϵ0ˆy⋅dyˆy | ΔV=−∫d0−σϵ0ˆy⋅dyˆy | ||
Since ˆy⋅ˆy=1 (because the two vectors are parallel), the integral becomes: | Since ˆy⋅ˆy=1 (because the two vectors are parallel), the integral becomes: | ||
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//__This expression is only true for parallel plate capacitors__//, | //__This expression is only true for parallel plate capacitors__//, | ||
- | === Dielectrics === | + | ==== Dielectrics |
- | What would change about our capacitance expression if we added a dielectric insulator in between the plates? As we talked about before, adding a dielectric allows more charge to pushed onto the plates of the capacitor for a certain voltage. If we think about that in terms of the capacitance equation Q=CΔV, this means that the proportionality constant (really the capacitance) should increase if a dielectric is added. | + | What would change about our capacitance expression if we added a dielectric insulator in between the plates? |
This is generally achieved by adding a constant to the capacitance called the dielectric constant. This constant is really a " | This is generally achieved by adding a constant to the capacitance called the dielectric constant. This constant is really a " | ||
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==== Examples ==== | ==== Examples ==== | ||
- | [[: | + | * [[: |
- | + | * Video Example: Energy Stored in a Parallel Plate Capacitor | |
- | [[: | + | |
+ | * Video Example: Finding the Capacitance of a Cylindrical Capacitor | ||
+ | {{youtube> | ||
+ | {{youtube> |