Group 7 Kaitlyn O. & Melvin K. & Chad C. PHYS212 Week 6: Capacitance Goal: To understand how capacitance varies with geometry Theory: Capacitance is defined as a proportionality constant between the amount of charge stored on a set of plates and the voltage across those plates. (Note that there can be alternate definitions, depending on the application.) The governing equation is: 1) q = CV where q is the stored charge, V is the voltage across the plates, and C is the capacitance. Understanding that for a single (large) plate holding charge, the electric field is given by where σ is the areal charge density (= q/A), it is possible to derive the field of two parallel plates of opposite charge by adding the vector fields together. Note that to a good approximation with real capacitors between the plates, the fields will add while outside the plates the fields will cancel. The magnitude of the resulting field between plates is: The relationship between electric field and voltage for a parallel plate capacitor can be found by determination of the work needed to move a positive test charge against the electric field across the plates of the capacitor.
where in the above line we have set the magnitude of the force equal to qE and resolved the unit vectors. The force is in the negative y direction whereas the motion against the field goes in the positive y direction. This integral evaluates to: The voltage across the plates is defined by the potential energy per unit charge. Since potential energy is the negative of work, we have: Substitution of this expression for V into equation 1) gives us: 8) q=CEd which becomes: after substituting for E from equation 3). Elimination of q and rearrangement results in the expression for the capacitance of a parallel plate capacitor: where d is the distance between the plates of the capacitor. This may be more generally written as: where κ is the dielectric constant of the material.
Part 1: Initial Calculations Capacitance of a charge geometry can be calculated in the following steps: a) Determine the electric field of the geometry using Gauss' law or Coulomb's law b) Determine the voltage (potential) from the electric field. Alternatively, in some cases it may be easier to calculate the voltage rather the E field. In such a case, step (a) may be skipped. c) Generally, the voltage is a function of charge and other stuff. For example if the voltage contains a charge density λ, this is charge over length ( λ = q/L). Rearrange your voltage equation to solve for charge in terms of voltage. Any variables that are not either charge (q) or voltage (V) must together comprise the capacitance (C) since q = CV.
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