Question

Perform the following computations. Display your answer in scientific notation. Three capacitors, \(8.26 \times 10^{-6}\) farad \((\mathrm{F}), 1.38 \times 10^{-7} \mathrm{F},\) and \(5.93 \times 10^{-5} \mathrm{F}\) are wired in parallel. Find the equivalent capacitance using the formula \(C=C_{1}+C_{2}+C_{3}\)

Short answer

The equivalent capacitance is \(6.69 \times 10^{-5} F\).

Step-by-step solution

Identify Capacitance Values

We have three capacitors with the following capacitances: \(C_1 = 8.26 \times 10^{-6} F\), \(C_2 = 1.38 \times 10^{-7} F\), and \(C_3 = 5.93 \times 10^{-5} F\).

Sum the Capacitances

Since the capacitors are in parallel, we sum their values to find the equivalent capacitance using the formula \(C = C_{1} + C_{2} + C_{3}\).

Calculate the Equivalent Capacitance

Add the values of \(C_1, C_2,\) and \(C_3\): \(C = (8.26 \times 10^{-6}) + (1.38 \times 10^{-7}) + (5.93 \times 10^{-5}) F\).

Convert to Scientific Notation

To ensure the answer is in scientific notation, all numbers must be converted to the form \(n \times 10^m\), and then summed: \(C = (0.0000826) + (0.000000138) + (0.000593) F\). Simplify the sum to get the equivalent capacitance in scientific notation.

Key concepts

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It uses powers of ten to compress the number into a more manageable form, making it easier to read, write, and calculate with these numbers, especially when dealing with measurements in physics and engineering.

In science and engineering, we often work with quantities that can be enormously large, such as the distance to galaxies, or incredibly small, such as the charge of an electron. Writing out such numbers in full would be impractical and prone to error. To handle this, scientific notation comes in handy. For instance, the number 0.0000826 can be written as \(8.26 \times 10^{-5}\), signifying that the decimal point has been moved five places to the left.

When calculating with scientific notation, it's important to keep track of the exponents. If you are adding or subtracting numbers in scientific notation, they must first be expressed with the same exponent. On the other hand, when multiplying or dividing, you can directly manipulate the exponents. Mastering scientific notation is essential for working with electronic components such as capacitors in circuits where precision is key.

Capacitors in Parallel

A common configuration in electronics is having capacitors in parallel. When capacitors are connected in parallel, their plates are connected to the same two points in a circuit, which makes the potential difference across them the same. This setup has the advantage of increasing the total capacitance available in the circuit.

The principle behind this is that when capacitors are in parallel, the total available plate area for storing charge increases, without increasing the voltage across the capacitors. This is critical because the capacitance is directly proportional to the plate area and inversely proportional to the distance between the plates, as given by the formula \(C = \epsilon_0 \frac{A}{d}\), where \(C\) is the capacitance, \(\epsilon_0\) the permittivity of free space, \(A\) the area of the plates, and \(d\) the distance between them.

In terms of calculating the total capacitance, you simply add up the capacitance values of each individual capacitor. This is a straightforward method that allows for quick modification of a circuit's overall capacitance without needing to replace every component, providing flexibility in design and debugging.

Summing Capacitances

When dealing with capacitors in parallel, summing their capacitances is a fundamental concept. The equivalent capacitance of several capacitors in parallel is the sum of their individual capacitances. Mathematically, if you have capacitors with capacitances \(C_1, C_2,..., C_n\), the total capacitance \(C\) is given by:\[C = C_1 + C_2 + ... + C_n\].

For example, consider you have three capacitors with the values \(C_1 = 8.26 \times 10^{-6} F\), \(C_2 = 1.38 \times 10^{-7} F\), and \(C_3 = 5.93 \times 10^{-5} F\). To find the total capacitance, you add these values together. The addition should be done carefully, keeping track of the exponents if the values are in scientific notation to ensure accuracy.

To confirm the result is in scientific notation, you may need to adjust the final sum so it has one non-zero digit to the left of the decimal point. This is crucial in electronics and physics, where standardized units and clear notation prevent misinterpretation and ensure consistency across technical communications.