Estimate whether corona discharge is likely to take place from the surface of an oil drop of radius \(5 \mu \mathrm{m}\) carrying about 100 electronic charges.

The electric field is a fundamental concept in physics, and it describes the force per unit charge experienced by a stationary charge in the vicinity of other charges. It's a vector field, meaning it has both magnitude and direction. Imagine the oil drop mentioned in the exercise as a point charge for simplicity.

The electric field (\(E\)) created by a charge is strongest close to the charge and weakens with increasing distance. Mathematically, it’s expressed by the equation \(E = \frac{kQ}{r^2}\), which shows that the field strength (\(E\)) is proportional to the charge (\(Q\)) and inversely proportional to the square of the distance (\(r^2\)) from the charge.

Applying the Concept to the Exercise

By utilizing this formula, we find that the electric field at the surface of the oil drop is significant. The calculated field strength of approximately \(5.73 \times 10^6 \, \mathrm{V/m}\) indicates a strong likelihood of an electric disturbance in the surrounding air—a precursor to corona discharge.

Dielectric strength refers to the maximum electric field that a material can withstand without experiencing electrical breakdown. It is a measure of a material's capability to prevent the flow of electric current, essentially acting as an insulator. For air, this threshold is typically about \(3 \times 10^6 \, \mathrm{V/m}\).

When the electric field exceeds this limit, the material (in this case, air) can no longer act as an insulating barrier, leading to a phenomenon known as electrical breakdown or corona discharge. This electric breakdown can allow charged particles to pass through, creating a conductive pathway through the previously insulating medium.

Connection to the Oil Drop Exercise

In our exercise, the electric field at the surface of the oil drop is greater than the dielectric strength of air. This contrast between the calculated electric field and air's threshold predicts that the oil drop can induce a corona discharge, where air's insulating properties are compromised, and it begins to conduct electricity.

Coulomb's law is a quantitative expression describing the force between two point charges. It states that the electric force (\(F\)) between two point charges is directly proportional to the product of their charges (\(Q_1, Q_2\)), and inversely proportional to the square of the distance (\(r^2\)) between them, encapsulated by the equation \(F = k \frac{Q_1 \cdot Q_2 }{r^2}\), where \(k\) is Coulomb's constant.

Coulomb's law is fundamental in understanding electrostatic interactions and plays a critical role in calculating electric fields. It provides a basis for why charges either repel or attract each other with a force that decreases as they move farther apart.

Relevance to Corona Discharge

If we apply Coulomb's law to the charges on the oil drop, we find that these charges exert forces on nearby air molecules. As the electric field intensity escalates due to the proximity of multiple charges, as on our oil drop, the chance of overcoming air's dielectric strength increases, leading to corona discharge, and Coulomb's law helps us compute the intensity of these interactions, integral to determining the electric field strength around the drop.