# RC to move have moved, and current stops

RCcircuit. A resistor–capacitor circuit (RCcircuit), or RC filter orRC network, is anelectric circuit composed of resistors and capacitors drivenby a voltage or current source.

A first order RC circuit iscomposed of one resistor and one capacitor and is thesimplest type of RC circuit.MAIN USAGE Theory: Capacitorsare energy storage devices. A capacitor stores energy in an electric field.When a potential is placed across a capacitor, the positive charges gather onthe side connected to the positive terminal of the battery, and the negativecollect charges on the other side. At some point all the charges that are freeto move have moved, and current stops flowing in the circuit. This capacity tohold charge is measured in farads, or more practically, microfarads. Considerthe series RC circuit shown below:Applyingthe loop law to this circuit, EMF – iR – q/C = 0, where is the terminal voltageof the battery, i (lower case because it varies with time) times R is thevoltage drop across the resistor, and q (lower case because it varies withtime) divided by C is the potential cross the capacitor.

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The battery sourcevoltage is the EMF. Rearranging:EMF = iR + q/Ceq. 1Wecannot solve this equation as it stands because both i and q are unknowns.However, they are related through calculus by the equation:i = dq/dteq. 2The’d’ is much like delta in the 100 courses, only much smaller. Through muchhocus pocus we substitute equation 2 into equation 1 and (gasp!) integrate. Thisyields, upon rearranging:q = EMF C(1- e(-t/RC))eq.

3Rememberingthat V = q/C, and letting EMF = Vo, we have an expression for thevoltage across the capacitor at any arbitrary time t:V = Vo(1 – e(-t/RC))eq. 4Thequantity RC iscalled the time constant. This is why we have a resistor in the circuit. Thinkwhat would happen if R = 0.

V = Vo at all times!Whenone disconnects the battery from the circuit, the capacitor will dischargeaccording to a similar equation. This is easily found. With the batterydisconnected, the EMF in the circuit = 0. Solving the differential equationwith EMF = 0 yields:q = qoe(-t/RC)eq. 5Now,suppose we connect the resistor in parallel with the capacitor:Atfirst, all the free charges move rapidly to their respective plates withessentially none left to flow through the resistor. As the capacitor becomescharged, less current flows through it and more is left over for the resistor.At some point the capacitor will have the maximum charge allowable, and thecurrent through the resistor will be more or less constant. If we nowdisconnect the battery, the charges stored on the capacitor are free to movethrough the resistor toward each other (remember that like charges repel, andthat they have been crammed up together on their respective plates).

Thecapacitor discharges through the resistor with the discharge rate dependentupon the size of the charge and the value of the resistor.·        Capacitors store electric charge·        When capacitors are charged using a voltage source in serieswith a resistor, the rate of change in capacitor voltage slows exponentially·        RC circuits are characterized by the RC product, called the timeconstant·        A capacitor in a simple RC circuit moves to 63% of thedifference between its current value and a step voltage applied to the circuitin one time constant·        A commonly accepted time for a capacitor to fully charge isequal to five time constants