When two capacitances are connected in series, show that the equivalent capacitance is given by 1 𝐶𝑒𝑞 = 1 𝐶1 + 1 𝐶2

To show that the equivalent capacitance when two capacitances are connected in series is given by 1/Ceq = 1/C1 + 1/C2, we can follow these steps:

Step 1: Start with the definition of capacitance. The capacitance of a capacitor is given by C = Q/V, where Q is the charge stored on the capacitor and V is the voltage across the capacitor.

Step 2: Consider two capacitors, C1 and C2, connected in series. This means that the positive terminal of C1 is connected to the negative terminal of C2, and the remaining terminals are connected to the circuit.

Step 3: When capacitors are connected in series, the same charge Q flows through each capacitor. This is because the charge cannot accumulate or be lost in the series connection.

Step 4: Let Q1 be the charge on capacitor C1 and Q2 be the charge on capacitor C2. Since the same charge flows through both capacitors, we have Q1 = Q2 = Q.

Step 5: Now, let's find the voltage across each capacitor. The total voltage across the series combination of capacitors is equal to the sum of the voltages across each capacitor. Let V1 be the voltage across capacitor C1 and V2 be the voltage across capacitor C2.

Step 6: Using the definition of capacitance, we can write V1 = Q1/C1 and V2 = Q2/C2.

Step 7: Since Q1 = Q2 = Q, we can rewrite the above equations as V1 = Q/C1 and V2 = Q/C2.

Step 8: The total voltage across the series combination is V = V1 + V2.

Step 9: Substituting the values of V1 and V2, we get V = Q/C1 + Q/C2.

Step 10: Now, let's find the equivalent capacitance Ceq. The equivalent capacitance is defined as the capacitance of a single capacitor that would store the same amount of charge Q when the same voltage V is applied.

Step 11: Using the definition of capacitance, we can write Ceq = Q/V.

Step 12: Substituting the value of Q from step 4 and the value of V from step 8, we get Ceq = Q/(Q/C1 + Q/C2).

Step 13: Simplifying the expression, we get Ceq = 1/(1/C1 + 1/C2).

Therefore, we have shown that the equivalent capacitance when two capacitances are connected in series is given by 1/Ceq = 1/C1 + 1/C2.