Problem 42 - University Physics with Modern Physics

Question

(a) How much excess charge must be placed on a copper sphere 25.0 \(\mathrm{cm}\) in diameter so that the potential of its center, relative to infinity, is 1.50 \(\mathrm{kV}\) ? (b) What is the potential of the sphere's surface relative to infinity?

Step-by-Step Solution

Answer

(a) 2.08×10⁻⁸ C (b) 1.50 kV

Step 1: Understand the problem

We need to find the excess charge on a copper sphere in order to achieve a certain electric potential difference from infinity. The sphere has a given diameter and we are provided with a potential relative to infinity. We also need to find the potential at its surface.

Step 2: Use the formula for electric potential

The electric potential \( V \) at the center and surface of a sphere due to its own charge \( Q \) is the same. The formula to use is: \[ V = \dfrac{kQ}{R} \] where \( k \) is Coulomb's constant \( 8.99 \times 10^9 \ \text{Nm}^2/\text{C}^2 \) and \( R \) is the radius of the sphere.

Step 3: Convert the diameter to radius (in meters)

The diameter of the sphere is given as \( 25.0 \text{ cm} \). Convert this to meters by dividing by 100, resulting in \( 0.25 \text{ m} \). Thus, the radius \( R \) is \( 0.125 \text{ m} \).

Step 4: Solve for the excess charge \( Q \)

Rearrange the formula for \( Q \):\[ Q = \dfrac{VR}{k} \]Substitute the given potential \( V = 1.50 \times 10^3 \text{ V} \), \( R = 0.125 \text{ m} \), and \( k = 8.99 \times 10^9 \ \text{Nm}^2/\text{C}^2 \) into the equation:\[ Q = \dfrac{(1.50 \times 10^3)(0.125)}{8.99 \times 10^9} \text{ C} \]Calculate the value to get the charge.

Step 5: Compute the excess charge

Perform the calculation:\[ Q = \dfrac{(1.50 \times 10^3)(0.125)}{8.99 \times 10^9} = 2.08 \times 10^{-8} \text{ C} \]Thus, the excess charge needed is \( 2.08 \times 10^{-8} \text{ C} \).

Step 6: Determine potential at sphere's surface

The potential at the surface of a conducting sphere due to its charge is the same as at its center because the potential at any point inside a conductor in electrostatic equilibrium is constant. Therefore, the potential of the sphere's surface relative to infinity is also \( 1.50 \text{ kV} \).

Key Concepts

Excess Charge

Excess charge refers to the additional electric charge that is placed on an object, beyond its natural or neutral state.
In the context of a copper sphere, this excess charge is the amount of charge required to create a specific electric potential.
When determining excess charge, the key is to understand how additional electrons (negative charge) or protons (positive charge) change the object's electric properties.

• Excess charge can be thought of as the "surplus" charge needed to achieve a certain potential.
• It can either be positive or negative, depending on whether electrons are added or removed.
• In our exercise, we calculated the specific amount of excess charge necessary for a potential of 1.50 kV.

The formula used to find this charge involves the sphere's radius and known physical constants like Coulomb's constant.

Conducting Sphere

A conducting sphere is a spherical object made of conductive material, such as copper, which allows electrons to move freely.
When a charge is placed on a conductor, it distributes evenly over the surface of the sphere due to the movement of electrons. This even distribution is crucial for maintaining consistent potential across the sphere.

• Conducting means the material allows free movement of electrons, leading to uniform charge distribution.
• The potential of a conducting sphere depends on its excess charge and its size (radius).
• The electric field inside a conductor in equilibrium is zero, making the potential constant throughout.

In the exercise, both the center and the surface of the sphere have the same potential, emphasizing the uniform nature of conductors.

Coulomb's Law

Coulomb's law is a fundamental principle that describes the force between two charged objects. The formula shows that this force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them:\[ F = k \frac{{|q_1 q_2|}}{{r^2}} \]

• This law helps in understanding how charges interact at certain distances.
• In our problem, while calculating potential, we used a related formula derived from Coulomb's principles.
• The constant \( k \) in the formula is crucial for calculations involving electric fields and potentials.

Understanding this law is vital for grasping how charges generate electric potential and influence the environment around them.

Electric Potential Difference

Electric potential difference is essentially the work needed to move a charge from one point to another in an electric field.
This potential is typically measured in volts and is what we sought to determine in the exercise.

• The potential difference from infinity to a point is the work done moving a charge from an area with no influences to the desired location.
• For a sphere, the potential at the surface is the same as the center, due to its conducting properties.
• It's crucial for understanding how energy is used and stored in electric systems, like batteries and capacitors.

Calculating the potential at various points of the sphere helps explain how electric fields and potentials operate around conductors.