Underlying our description of the behavior of capacitors that we see in the main article is some mathematics. Involved in this math is some basic calculus. Calculus is merely the math of movement and change. It was developed by Newton and Leibniz to help them describe how physical objects moved and changed, usually with time. In electricity, we deal with moving electrons, so this math of movement finds another interesting application. Even though some of the symbols below may seem to be new, the underlying equations are quite simple. They use multiplication, division, and basic algebra. The only difference is that sometimes we are using descriptions of movement, or changes in quantity, rather than fixed values. Equation 1: The voltage (V) between the plates is a function of the total number of charges and the number of charges on the plate. We’ll call that “Q”. As algebra, this sentence is written: V = mxQ There’s nothing magical about the letters “Q” or “m,” they are just convenient placeholders, with Q for the number of charges, and m to tell us how many charges are added based on V. Equation 2: Next we have to define current (I). Current is equal to the rate of electron movement. That is to say that current tells how many electrons pass a point every second. This is electrons/second. Because this a rate, we can write it in calculus terms: I = dQ/dt Or in English as: “Current (I) equals the rate of flow of charge (dQ) per time (dt).” Equation 3: We can also write Equation 1 from the rate of change standpoint, again using calculus notation: DV/dt = m x dQ/dt In English: “The change in voltage (dV) versus time (dt) equals m times the change in charge (dQ) versus time (dt).” Equation 4: Now look at Equation 2 and see that it is similar to the right side of Equation 3. We can therefore substitute in using algebra and get: DV/dt = mxI In English: “The rate of change of voltage (dV) versus time (dt) equals m times current (I).” Equation 4 is a math way of saying: “the rate that the voltage is being built up between the capacitor plates is proportional to the rate that charge is being added to the plate.” This is exactly one of the written conclusions from the article. Equation 5: Now the magic of calculus can come into play. Let’s imagine we have a simple oscillating signal defined by a function: F(X, t) = A x sin(Xxt) In English: “The function of the signal in time (t) equals the sine of (t) times the frequency (X), and then times the signal’s amplitude (A)” Equation 6: If we plug in f(t) for the voltage (V) in Equation 4, we get: D(f(X, t))/dt = mxI In English: “The rate of change of a periodic voltage (V) function of frequency (X) is proportional to the current.” Equation 7: Using some calculus details that we won’t go into here, we can solve Equation 6 for I. The solution of I ends up being: I(X, t) = A x X x (cos(X x t)) There are two key takeaways from Equation 7. First is that the capacitor changes the phase. This is demonstrated by the cosine term. Sine and cosine both step through the same range of values, but they have a phase shift between them. The second takeaway is that the current is small when frequency is low, or A x X is small. Alternatively, the current is large when frequency is high (i.e., A x X is large). This is a mathematical representation of the filter effect we see from capacitors! They block current at low frequencies (i.e. when A x X is small), but allow current at high frequencies (i.e., when A x X is large).