Problem 29 - Physics

Question

Two equipotential surfaces surround \(\mathrm{a}+1.50 \times 10^{-8}-\mathrm{C}\) point charge. How far is the 190-V surface from the 75.0 -V surface?

Step-by-Step Solution

Answer

The distance is approximately 1.09 meters.

Step 1: Understand the Concept of Equipotential Surfaces

Equipotential surfaces are surfaces where the electric potential is constant. The difference between these surfaces indicates a change in potential energy. In this problem, we are dealing with two surfaces at potentials of 190 V and 75 V.

Step 2: Recall the Formula for Electric Potential Around a Point Charge

The electric potential (V) at a distance (r) from a point charge (q) is given by the formula: \[V = \frac{k \cdot q}{r}\]where \(k\) is the Coulomb's constant (\(8.99 \times 10^9 \text{ N m}^2/\text{C}^2\)) and \(q\) is the point charge.

Step 3: Express Distances Based on Potentials

Let \(r_1\) be the distance to the 190-V surface and \(r_2\) be the distance to the 75.0-V surface. Using the electric potential formula, we have:\[190 = \frac{k \cdot a}{r_1}\]\[75 = \frac{k \cdot a}{r_2}\]

Step 4: Relate the Two Equations

From the equations derived, solve for \(r_1\) and \(r_2\):\[r_1 = \frac{k \cdot a}{190}\]\[r_2 = \frac{k \cdot a}{75}\]The difference \(r_2 - r_1\) will give the distance between the two surfaces.

Step 5: Calculate the Distance Between the Surfaces

Substitute \(a = 1.50 \times 10^{-8} \,\text{C}\) and \(k = 8.99 \times 10^9 \,\text{N m}^2/\text{C}^2\) into the equations:\[r_1 = \frac{8.99 \times 10^9 \,\text{N m}^2/\text{C}^2 \times 1.50 \times 10^{-8} \,\text{C}}{190} \approx 0.7105 \,\text{m}\]\[r_2 = \frac{8.99 \times 10^9 \,\text{N m}^2/\text{C}^2 \times 1.50 \times 10^{-8} \,\text{C}}{75} \approx 1.798 \,\text{m}\]Thus, the distance between the two surfaces is:\[1.798 \,\text{m} - 0.7105 \,\text{m} = 1.0875 \,\text{m}\]

Step 6: Conclusion

Therefore, the 190-V equipotential surface is approximately 1.09 meters from the 75.0-V equipotential surface.

Key Concepts

Equipotential Surfaces

Equipotential surfaces are fascinating and essential concepts in electrostatics. These surfaces represent regions in an electric field where every point has the same electric potential. Imagine them as invisible map contours but in the realm of electric fields. Along these surfaces, the potential energy does not change. Consequently, no work is done when moving a charge along an equipotential surface.

• They are always perpendicular to electric field lines.
• In the case of a point charge, these surfaces are spherical shells centered around the charge.
• The potential difference between any two equipotential surfaces gives insight into how much work would be needed to move a charge between them.

Recognizing equipotential surfaces helps to understand the spatial distribution of electric potential in a given field.

Electric Potential

Electric potential is a critical concept in understanding electric fields. It refers to the amount of electric potential energy per unit charge at a specific point in the field. It is often thought of as the energy landscape for charges within the field.

• Measured in volts (V), a scalar quantity.
• Describes the electric condition of an object placed in the field.
• Provides insight into how an electric field will influence a test charge.

The electric potential at a point is determined by the configuration of all local charges and is crucial for calculating potential energy differences, which are important in circuits and systems using electric power.

Coulomb's Constant

Coulomb's constant, often represented as \(k\), is a fundamental physical constant used in electrostatics. It ensures that we can quantify the amount of electrostatic force or electric potential exerted by a charge over a distance.

• The constant value is \(8.99 \times 10^9 \, \text{N m}^2/\text{C}^2\).
• It signifies the strength of the electrostatic force within a vacuum.
• Used in formulas like the electric potential formula \(V = \frac{k \cdot q}{r}\).

Coulomb's constant is the bridge between the charge amount, distance, and resulting electromagnetic interaction, making it vital for calculations involving charges and potentials.

Point Charge

In electrostatics, a point charge is an idealized model representing a charged object with negligible physical dimensions compared to the problem's scale. Through this simplification:

• All the charge is considered to be located at a single point.
• Commonly used in theoretical models and problems to simplify calculations.
• Allows the use of straightforward equations to determine electric potential and field strength.

By treating charges as point charges, the complex electric phenomena they generate can be more easily understood and mathematically modeled.

Potential Difference

Potential difference, also known as voltage, is the measure of the difference in electric potential between two points. It's the "push" or "pull" that drives electric charges to move through a conductor.

• Measured in volts.
• Represents the work needed per unit charge to move a charge between two points against an electric field.
• In the exercise, it signifies the difference in potential energy between the two equipotential surfaces.

Understanding potential difference is essential for analyzing electric circuits, as it dictates the current flow and energy distribution within the system.