You're developing a switch for high-voltage power lines. The smallest part in your design is a 5.0 -cm-diameter metal sphere. What do you specify for the maximum potential on your switch if the electric field at the sphere's surface isn't to exceed the 3 -MV/m breakdown field of air?

Understanding electric potential is crucial when dealing with high-voltage systems, such as the switch for power lines mentioned in our exercise. Electric potential, often simply called voltage, is a measure of the electric potential energy per unit charge at a point in a static electric field. It's comparable to the height of a hill in a gravitational field: the higher you go, the more potential energy you have.

When discussing electric potential in a high-voltage switch design, you are essentially considering the 'electric height' from which electrons can 'fall' or be driven through the circuit. In the case of our metal sphere, the electric potential is the maximum energy that a charge would have if positioned on the sphere's surface. Mathematically speaking, it's a matter of measuring how much work an external agent would need to do to move a charge from a reference point (often infinitely far away) to the present point, without any acceleration.

To calculate the electric potential around our sphere, knowing its radius and the maximum electric field strength, we could use the formula mentioned in the solution, which is a simplified version of the more general equation, acknowledging that work is needed to move a charge against the electric field.

Electric field strength is a fundamental concept in understanding how electrical forces are transmitted through space. It is defined as the force experienced by a positive test charge at a given point divided by the magnitude of the charge itself. The unit for electric field strength is volts per meter (V/m). A strong electric field can exert substantial electromotive force on charged particles, such as electrons or ions.

In our switch design example, electric field strength refers to how intensely the electrical force is being exerted per unit charge at the metal sphere's surface. The formula used, \(E = \frac{V}{r}\), elegantly illustrates that electric field strength diminishes with distance (\(r\)) while being directly proportional to the electric potential (\(V\)). It's this field that can cause electrical breakdown in the surrounding air if it becomes too intense, which links directly to our next topic of breakdown field strength.

The breakdown field of air is a critical parameter in electrical engineering and high-voltage applications. It specifies the electric field strength at which air can no longer insulate electrical components, causing it to ionize, or become conductive. This typically results in unwanted electrical discharge or arcing -- similar to a miniature lightning strike.

The specified breakdown strength for air at standard conditions is about 3 megavolts per meter (MV/m). It's the maximum electric field that air can withstand before becoming electrically conductive. When we design high-voltage equipment we must ensure that the electric field does not surpass this limit; otherwise, the air surrounding the equipment could ionize, leading to potential damage or failure.

In the solution to the problem, this breakdown field strength is used to determine the maximum potential the metal sphere can handle without causing the surrounding air to ionize. By using the known breakdown field strength and the dimensions of the sphere, we can calculate the maximum safe electric potential, which ensures the designed switch operates within safe, non-ionizing conditions.