Problem 31 - Physics

Question

When you walk across a rug on a dry day, your body can become electrified, and its electric potential can change. When the potential becomes large enough, a spark of negative charges can jump between your hand and a metal surface. A spark occurs when the electric field strength created by the charges on your body reaches the dielectric strength of the air. The dielectric strength of the air is \(3.0 \times 10^{6} \mathrm{~N} / \mathrm{C}\) and is the electric field strength at which the air suffers electrical breakdown. Suppose a spark \(3.0 \mathrm{~mm}\) long jumps between your hand and a metal doorknob. Assuming that the electric field is uniform, find the potential difference \(\left(V_{\mathrm{knob}}-V_{\text {hand }}\right)\) between your hand and the doorknob.

Step-by-Step Solution

Answer

The potential difference is 9000 V.

Step 1: Understand the Problem

We need to find the potential difference between the hand and the doorknob when a 3.0 mm spark jumps. We know the dielectric strength of air is given as \(3.0 \times 10^{6} \mathrm{~N} / \mathrm{C}\), which is the electric field strength, \(E\).

Step 2: Relate Electric Field to Potential Difference

The potential difference \(V\) between two points separated by a distance \(d\) in a uniform electric field \(E\) is given by the formula \(V = E \times d\). In this problem, \(E = 3.0 \times 10^{6} \mathrm{~N} / \mathrm{C}\) and \(d = 3.0 \mathrm{~mm} = 3.0 \times 10^{-3} \mathrm{~m}\).

Step 3: Calculate the Potential Difference

Using the formula \(V = E \times d\), substitute the given values: \[V = 3.0 \times 10^{6} \mathrm{~N} / \mathrm{C} \times 3.0 \times 10^{-3} \mathrm{~m}\] Calculate the result.

Step 4: Simplify the Calculation

Multiply the electric field and the distance: \(V = 3.0 \times 3.0 = 9.0\) with powers of 10: \(10^{6} \times 10^{-3} = 10^3\). Combine these to get the potential difference: \[V = 9.0 \times 10^{3} \mathrm{~V} = 9000 \mathrm{~V}\].

Key Concepts

Dielectric Strength

The dielectric strength of a material, like air, refers to the maximum electric field that the material can withstand without breaking down. This breakdown occurs when the insulating properties of the material fail and it becomes conductive, allowing charges to move freely.

In the context of the given problem, the dielectric strength of air is significant because it sets the limit for when a spark can jump in air. When an electric field surpasses this threshold, known to be approximately \(3.0 \times 10^{6} \text{ N/C}\) for air, a spark or discharge occurs, indicating an electrical breakdown.

• The concept of dielectric strength is crucial in designing electrical insulations, ensuring they can endure electric fields without failing.
• Understanding this property helps in applications where preventing sparks or arcs is necessary for safety.

Electric Field

The electric field is a vector quantity that represents the force per unit charge at a particular point in space. It dictates how charges interact within a field and is crucial for understanding electrostatic interactions.

In simpler terms, it tells us how a charged object would affect other charges that come into its proximity. The magnitude and direction of the electric field depend on the source charges' properties and the distance from them.

• Electric fields are foundational in electromagnetism, a branch of physics that studies electric and magnetic fields.
• They play a role in many everyday phenomena, such as the static electricity you experience when touching a metal object after walking on a carpet.

Potential Difference

Potential difference, often referred to as voltage, is the difference in electric potential between two points. It is the work required to move a unit charge from one point to another against the electric field.

In the exercise at hand, the potential difference between your hand and a doorknob is the energy per charge that causes a spark to form. Calculating this potential difference helps us understand the energy dynamics involved when charges move through a medium like air.

• The potential difference is directly proportional to the electric field strength and the distance over which the field acts, supporting the relationship \(V = E \times d\).
• Understanding voltage is essential for electrical circuits, battery technologies, and safety standards in electrical engineering.

Uniform Electric Field

A uniform electric field is one where the field strength is constant at every point between two charged plates or surfaces. This means that the force experienced by a charge in this field doesn't change, leading to a predictable behavior of charges.

In the scenario described in the exercise, the assumption of a uniform electric field between your hand and a doorknob simplifies the calculations. This is because the potential difference becomes simply the product of the electric field strength and the distance.

• Uniform electric fields are idealized forms that help to better understand concepts in electrostatics, especially in classroom scenarios.
• They make calculations manageable and are the basis for understanding more complex, real-world electric fields.

Electrical Breakdown

Electrical breakdown occurs when an insulating material becomes conductive, often resulting in a spark or arc. This phenomenon is caused by an electric field so strong that it frees electrons from atoms, thereby turning the insulator into a conductor.

In practical terms, when the electric field in a material exceeds its dielectric strength, it can no longer insulate effectively, leading to electrical breakdown. In the given problem, this is what causes the spark when touching a metal doorknob.

• Understanding electrical breakdown is crucial for designing safe electrical systems to prevent uncontrolled discharges.
• It emphasizes the importance of selecting proper insulating materials to withstand expected electric fields in various devices.