184_notes:cap_in_cir

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184_notes:cap_in_cir [2019/01/04 03:02] – dmcpadden184_notes:cap_in_cir [2021/06/15 00:35] (current) – schram45
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 Section 19.1 in Matter and Interactions (4th edition) Section 19.1 in Matter and Interactions (4th edition)
  
-[[184_notes:r_series|Next Page: Resistors in Series]]+/*[[184_notes:r_series|Next Page: Resistors in Series]]
  
-[[184_notes:cap_charging|Previous Page: Charging and Discharging Capacitors]]+[[184_notes:cap_charging|Previous Page: Charging and Discharging Capacitors]]*/
  
 ===== 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/volt (C/V), which is called a farad (named after [[https://en.wikipedia.org/wiki/Michael_Faraday|Michael Faraday]]) and abbreviated with "F". A 1 F capacitor is a extremely large capacitor - large in physical size and also takes a long time to charge/discharge. A typical capacitor in electronic capacitors is on the order of nanofarads (10−9) to picofarads (10−12).+The units of capacitance are then **coulombs/volt (C/V), which is called a farad** (named after [[https://en.wikipedia.org/wiki/Michael_Faraday|Michael Faraday]]) and abbreviated with "F". A 1 F capacitor is an extremely large capacitor - large in both in physical size and the amount of time it takes to charge/discharge. A typical capacitor in electronic capacitors is on the order of nanofarads (10−9) to picofarads (10−12).
  
 ==== Finding Capacitance ==== ==== Finding Capacitance ====
 We can actually find an expression for capacitance specifically for parallel plates using the [[184_notes:pc_vefu|relationship between electric field and electric potential]] and the definition of capacitance above. //__Assume we start with a pair of parallel plates__//, which have an area A, a charge of +/−Q, and are separated by a distance d. We can actually find an expression for capacitance specifically for parallel plates using the [[184_notes:pc_vefu|relationship between electric field and electric potential]] and the definition of capacitance above. //__Assume we start with a pair of parallel plates__//, which have an area A, a charge of +/−Q, and are separated by a distance d.
  
-[{{  184_notes:5b_diagram_solution.jpg?300|Electric field between two parallel plates}}]+[{{ :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 //__assuming that the charge density is uniform__// and the ϵ0 constant. We represent the charge density with a σ: 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 //__assuming that the charge density is uniform__// and the ϵ0 constant. We represent the charge density with a σ:
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 //__This expression is only true for parallel plate capacitors__//, but shows an important result. **The capacitance of the plates only depends on the size/shape of plates (area) and how far apart the plates are**, not on the amount of charge on the plates or the potential difference.  //__This expression is only true for parallel plate capacitors__//, but shows an important result. **The capacitance of the plates only depends on the size/shape of plates (area) and how far apart the plates are**, not on the amount of charge on the plates or the potential difference. 
  
-=== Dielectrics ===+==== Dielectrics ====
 What would change about our capacitance expression if we added a dielectric insulator in between the plates? [[184_notes:cap_charging|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? [[184_notes:cap_charging|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.
  
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 ==== Examples ==== ==== Examples ====
-[[:184_notes:examples:Week7_energy_plate_capacitor|Energy Stored in a Parallel Plate Capacitor]] +  * [[:184_notes:examples:Week7_energy_plate_capacitor|Energy Stored in a Parallel Plate Capacitor]] 
- +    * Video Example: Energy Stored in a Parallel Plate Capacitor 
-[[:184_notes:examples:Week7_cylindrical_capacitor|Finding the Capacitance of a Cylindrical Capacitor]]+  * [[:184_notes:examples:Week7_cylindrical_capacitor|Finding the Capacitance of a Cylindrical Capacitor]] 
 +    * Video Example: Finding the Capacitance of a Cylindrical Capacitor 
 +{{youtube>AAGXqz2JMbc?large}} 
 +{{youtube>vp_VWtwdMkE?large}}
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