4/28 Capacitors and Capacitive Circuits

Purpose: Today we continue the semester with the study of capacitors and capacitive circuits. A capacitor is a device that stores electric charge and electrical potential energy and capacitance is a measure of the ability of a device to store charge per unit of voltage applied across the device. We will analyze how measuring capacitance depends on area and/ or separation, and how capacitance works and is calculated in a series and parallel circuit.

Definition: Capacitors and Capacitive Circuits

Any two conductors separated by an insulator can be electrically charged so that one conductor has a positive charge and the other conductor has an equal amount of negative charge;  such an arrangement is called a capacitor.  A capacitor can be made up of two strange-shaped blobs of metal or it can have any number of regular symmetric shapes such as that of one hollow sphere inside another, or one hollow rod inside another.

We began class today with a quiz on a circuit which included, current, resistance, and voltage. What we did in order to solve this problem was to use Kirchhoff's Law. First we assigned currents, second applied the loop rule, last designated the direction of the current. After finding all three currents, we were asking to find power using P=I^2*R we calculated the power to be 0.42W and 0.7956W.

Capacitance Measurements for parallel plates

Capacitance is defined as a measure of the amount of net or excess charge on either one of the conductors per unit voltage, given by the equation: C=Q/V

Activity:  Measuring How Capacitance Depends on Area or on Separation

a. Devise a way to measure how the capacitance depends on either the foil area or on the separation between foil sheets.  If you hold the area constant and vary separation, record the dimensions of the foil so you can calculate the area.  Alternatively, if you hold the distance constant, record its value.  Take at least five data points in either case.  Describe your method and then create a data table with proper units and display a graph of the results.

Pictured below is our next experiment. What we did was we got a book and two pieces of aluminum, we then place each aluminum piece in different pages of the book. We connected the multimeter to the ends of each aluminum piece with alligator clips, using logger pro we were able to measure its capacitance, which depends on its distance. Pictured below is a chart with the values we attained and the distance of each.

Pictured below are our graphs from logger pro for capacitance. We found that the capacitance was not linear. This is because the capacitance is directly proportional to the area and inversely proportional to the distance.

Energy Storage: 

The work it takes to charge up a capacitor is equal to the electrical energy stored in a capacitor. 

The work it takes to move a differential charge dq through a voltage V is V dq. Integrating from when there is no charge on the capacitor until the capacitor is fully charge with a charge Q we find:

The work it takes to charge up a capacitor is equal to the electrical energy stored in a capacitor. The work it takes to move a differential charge dq through a voltage V is Vdq. Integration from when there is no charge on the capacitor until the capacitor is fully charge with a charge Q is equal to 1/2*Q^2/V (pictured below).

Activity:  Derivation of Capacitance vs. A and d (pictured below)

Capacitance can now involve electric field.

DIELECTRICS

We were given the following information about dielectrics: 

Most Capacitors have nonconducting material - a dielectric - between their plates. 

Dielectric materials increase the capacitance C of a capacitor; usually by several orders of magnitude compared to air filled capacitors. 

Dielectrics help keep the electrical plates separated without electrical contact. This also allows the separation of the plates to be reduced and thus increasing the capacitance. 

When a dielectric is placed between the plates of a capacitor, the permittivity of the space between the capacitors change from that of empty space ε_o to that of the dielectric material ε. 

For practical reasons it is more useful to define and use dielectric constant κ which is the ratio of ε_o over ε when determining the change in capacitance due to the presence of a dielectric. 

The Dielectric Strength is the maximum Electric Field that the material can withstand before the material breaks down as an insulator and permits current to flow through the material. 

Capacitors in series and in parallel

 We continues our studies by conducting a couple of capacitance problems, and what we found was that when in series total capacitance is the sum of its inverses; when in parallel its total is the sum of all capacitance. In summary, capacitance is the opposite of resistance when calculating the total.

Professor mason then gave us a few capacitance problems to solve. The one pictured below has capacitors in parallel and in series. We first calculated the ones in series then the one in parallel to each other.

This next example was a little different. Here we are given a battery in the middle of the circuit and we are asked to find the voltage. We solve for capacitance in series then in parallel from there we used Q=CV to find Q(charge) then V(voltage) for each.

And finally, the last part of class we were asked to find the work on a battery. We used change in potential energy (work) equal to 1/2*C*V^2, using the voltages and capacitance from the pervious example, we were able to calculate the total work done (pictured below).