Capacitance and Dielectrics

• Storing Charge
• Connect two conducting plates to battery (source of potential difference)
• Charge will move
• Equilibrium
• DV _plates =D V_battery
• Plate connected to negative terminal
• negative charge
• Plate connected to positive terminal
• positive charge
• Capacitor
• stores charge
• stores energy
• The larger the voltage the more charge can be stored
• Capacitance
• ratio of stored charge to applied voltage
• C=Q/V
• units
• C/V = Farad = F (in honor of Michael Faraday)
• Capacitance is independent of Q and V, depends on geometry of the capacitor
• Parallel Plate Capacitor
• Area (A), separation of plates (d)
• Qualitative Estimate for Capacitance
• depends on area
• larger area-more charge
• separation of plates
• closer together - more charge
• permittivity of free space
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• Quantitative Value for Capacitance
• The electric field between the plates is uniform and can be determined as follows
• the electric field due to the positive plate



• away from positive plate



• the electric field due to the negative plate



• toward negative plate or in same direction as that due to the positive plate
• the total electric field



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• the electric potential
• V = Ed



• C = Q/V



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• Other Shapes



• Examples



• Circuits
• Battery -
• Capacitor -



• Combining Capacitors
• replace combination by a single equivalent capacitance
• battery will contribute a given amount of charge
• *circuit - every point on wire is at same potential
• Parallel
• each capacitor is connected to the same two points
• potential difference across each capacitor is the same
• Finding the equivalent capacitance
  


• V₁ = V₂ = V_b
• Q₁ + Q₂ = Q_t
• Q_t = C_eqV
• C₁ V + C₂ V = C_eqV
• C₁ + C₂ = C_eq
• Capacitance increases
• Series
• capacitors are connected end to end
• each capacitor stores same charge (ideal)
• Finding the equivalent capacitance


• Q₁ = Q₂ = Q_t
• V₁ + V₂ = V_t
• Q_t = C_eqV
• Q/C₁ + Q/C₂ = Q/C_eq
• 1/C₁ + 1/C₂ = 1/C_eq
• Combination
• find series and parallel combinations
• replace with equivalent capacitance
• continue until the circuit has been reduced to one equivalent capacitance



• Examples
• Storing Energy
• stored energy = to work done to obtain configuration
• start with neutral plate
• no potential difference across plate
• bring in small amount of charge (DQ)
• no work done
• now potential difference exists across plates
• V = DQ/C
• bring in more charge
• DW = DQV
• V will depend on Q
• W = area under line

• area of triangle

• W = 1/2QV = 1/2CV² = 1/2 Q²/2C

• or if D q is too big

• dW = V dq

• work done to bring in infinitesimally small bit of charge
• total work



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• The energy stored is equal to the work done while obtaining the arrangement,
• an electric field exists between the plates, and we could think about the energy stored in terms of the electric field.

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• (A d) is just the volume between the plates, energy density



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• Examples