PHYS219 Fall semester 2014 Lecture 06: Capacitors Dimitrios Giannios Purdue University

The standard simple model Δ E inside is uniform to a good approximation. E outside is zero to a good approximation parallel plate capacitor filled with air -12 Note 1: εo=8.85×10 C2/(N•m2) Note 2: C depends on geometry Note 3: A is area of one plate 2

Relating V to E – general case Δ(PEE) + W=0 F F =qtest E F|| = qtest E|| initial q test Δs ΔW= F|| Δs W = ΣΔW = qtest ΣE|| Δs = -Δ(PEE) = -qtestΔV Q4 Q1 x final Q Q2 3 How V changes with Δ s gives quantitative information about how E varies with position

How is the potential difference ΔV related to E for the case of uniform E field? Two parallel charged plates separated by a distance d (the case of uniform E) FE = qtest E Δs W = FE Δs = qtest E Δs but W = - qtest ΔV Δ V = - E Δ s or (previous slide) The variation in electric potential with position is linked to the electric field!

Example Increasing electrostatic potential What is the magnitude and direction of the uniform electric field if the potential V is measured to be V1 = +15 V and V2 = +10 V at two points separated by 0.2 m? + ____ _ _ _ _ _ x

Example: Variation of potential in 1-dimension

Derivation for Capacitance of Parallel Plates Focus on positive plate First, let’s calculate the electric field http://phet.colorado.edu/sims/charges-and-fields/charges-and-fields_en.html

Derivation for Capacitance of Parallel Plates Focus on positive plate Gaussian surface qenclosed Gauss ' Law: E ,total  ε0

Derivation for Capacitance of Parallel Plates (More) q is charge on ONE plate Enet is field inside capacitor Negative plate Positive plate Gaussian surface d (separation between plates) ΔV (potential difference between plates) - - - - - - - - - - - - - - - - - - - - - - E neg In general, we also have

Nature’s capacitors Although relatively simple, the parallel plate capacitor can be used to provide simple models for capacitance in many situations, for example C1 C2

Energy Stored in Capacitors Example: If C=1μF and ΔV=1.5 V, how much energy is stored?

Real life capacitors Most real capacitors contain two metal “plates” separated by a thin insulating region Many times these plates are rolled into cylinders The region between the plates typically contains a material called a dielectric

The effect of dielectrics on capacitance (qualitative) 11 Uncharged capacitor for given charge at capacitor  Electric field reduced voltage reduced Capacitance increased Δ

κ = The effect of dielectrics (quantitative) Evacuum dielectric constant κ = Edielectric

Manipulating capacitance: Capacitors can be connected in one of two ways Series: +Q ‐Q =Q/(ΔV1+ΔV2) +Q ‐Q Parallel: =(Q1+Q2)/ΔV

What is the equivalent capacitance of this network? Example: What is the equivalent capacitance of this network? C/4

Example II: What is the equivalent capacitance? this network? 15.33 μF 3.33 μF Step 2: Step 1: Cequiv = C1 +C2 + C3 = 10 + 3.33 + 2 = 15.33