The potential difference across a cell membrane is \(65 \mathrm{mV}\). On the outside are \(1.5 \times 10^{6}\) singly ionized potassium atoms. Assuming an equal negative charge on the inside, find the membrane's capacitance.

When tackling problems in physics, such as calculating membrane capacitance, it's crucial to follow a systematic approach. Begin by identifying and listing the given values from the problem statement. In this membrane capacitance problem, the focus is on the potential difference and the quantity of charged particles. After listing the known information, proceed to connect these values using relevant physical concepts and equations. In this case, the relationship between charge, capacitance, and potential difference is essential.

Problem-solving in physics often involves breaking down a complex situation into fundamental principles. It's important to remember that each step in solving the exercise builds on the previous one, leading to a logical and coherent solution. Clarity of concepts, patient analysis, and methodical application of equations lead to successful problem resolution in physics.

Electrical capacitance is a measure of a system's ability to store electric charge. The SI unit of capacitance is the farad (F), named after the English physicist Michael Faraday. A capacitor is a component with this ability and consists of two conductors separated by an insulator. In our exercise, the cell membrane behaves similar to a capacitor.

In the equation for capacitance, defined as \( C = \frac{Q}{V} \), where \( C \) represents capacitance, \( Q \) is the total electric charge, and \( V \) is the potential difference across the conductors. Understanding this fundamental relationship aids in both understanding the concept of capacitance and in solving related problems.

Potential difference, or voltage, is a measure of the work done to move a charge from one point to another in an electric field. It's a critical concept in electromagnetism, and the potential difference across a cell membrane is vital for the conduction of signals in nerve and muscle cells.

Expressed in volts (V), potential difference in a circuit causes current to flow, and in our exercise, we use it to determine the membrane's capacitance. It acts as the driving force that causes electrons or ions to move, establishing an electric current. Recognizing the effects of potential difference on charge movement helps us comprehend how electric fields influence material properties.

The elementary charge, denoted by \( e \), is the smallest unit of electric charge that is considered indivisible in classical electromagnetism. Its value, approximately \( 1.602 \times 10^{-19} \mathrm{C} \), is fundamental to the structure of matter. In our calculation, we consider singly ionized potassium atoms, each carrying one elementary charge.

This concept is paramount when calculating the total charge carried by ions in the biological context of cell membranes. The understanding that each singly ionized atom donates one elementary charge aids significantly in determining the overall charge, which in turn is essential for calculating the capacitance of the membrane.