Question

A solid metal ball with a radius of \(3.00 \mathrm{~m}\) has a charge of \(4.00 \mathrm{mC}\). If the electric potential is zero far away from the ball, what is the electric potential at each of the following positions? a) at \(r=0 \mathrm{~m},\) the center of the ball b) at \(r=3.00 \mathrm{~m},\) on the surface of the ball c) at \(r=5.00 \mathrm{~m}\)

Short answer

Answer: The electric potentials at the given positions are: a) At the center of the ball, the electric potential is 0 V. b) On the surface of the ball, the electric potential is 1.199 x 10^7 V. c) At a distance of 5.00 m from the center of the ball, the electric potential is 7.190 x 10^6 V.

Step-by-step solution

Understand the concept of electric potential

Electric potential is the amount of work needed to move a test charge from a reference point to a specific point in an electric field without accelerating the test charge. The reference point can be chosen arbitrarily, and in this case, it is far away from the ball, where the electric potential is zero. The electric potential at any point is given by the formula \(V = \frac{kQ}{r}\), where \(V\) is the electric potential, \(k\) is the electrostatic constant (\(8.9875 \times 10^9 \mathrm{Nm^2/C^2}\)), \(Q\) is the charge of the source, and \(r\) is the distance between the point and the source.

Calculate the electric potential at \(r=0\mathrm{~m}\) (center of the ball)

At the center of the ball, \(r=0\mathrm{~m}\). Since the electric potential is the amount of work needed to move a test charge from a reference point to this point, and since the center of the ball is at zero distance from the source of charge, the electric field is uniform and doesn't change with the movement of a test charge. Therefore, the electric potential at the center of the ball, \(V_{center}\), is the same as that far away from the ball, which is zero. So, \(V_{center} = 0\mathrm{V}\).

Calculate the electric potential at \(r=3.00\mathrm{~m}\) (surface of the ball)

At the surface of the ball, \(r=3.00\mathrm{~m}\). We can use the formula for electric potential \(V = \frac{kQ}{r}\) to find the electric potential at the surface of the ball, \(V_{surface}\). Plugging in the values, we get \(V_{surface} = \frac{8.9875\times10^9\mathrm{Nm^2/C^2} \times 4.00\times10^{-3}\mathrm{C}}{3.00\mathrm{m}} = 1.199 \times 10^7\mathrm{V}\).

Calculate the electric potential at \(r=5.00\mathrm{~m}\)

At a distance of \(5.00\mathrm{~m}\) from the center of the ball, \(r=5.00\mathrm{~m}\). We can use the formula for electric potential \(V = \frac{kQ}{r}\) to find the electric potential at this point, \(V_{5m}\). Plugging in the values, we get \(V_{5m} = \frac{8.9875\times10^9\mathrm{Nm^2/C^2} \times 4.00\times10^{-3}\mathrm{C}}{5.00\mathrm{m}} = 7.190 \times 10^6\mathrm{V}\). In summary, the electric potentials at the given positions are: a) \(V_{center} = 0\mathrm{V}\) b) \(V_{surface} = 1.199 \times 10^7\mathrm{V}\) c) \(V_{5m} = 7.190 \times 10^6\mathrm{V}\)

Key concepts

Electrostatics

Electrostatics is the branch of physics that deals with the study of electric charges at rest. Unlike current electricity which involves moving charges, electrostatics focuses on the forces, fields, and potentials generated by stationary charges.

Electrostatic concepts are fundamental for understanding how electric charges interact and the resultant forces between them. The most common electrostatic phenomenon is the attraction and repulsion between charges: like charges repel each other, while unlike charges attract.

Key concepts in electrostatics include:

• **Coulomb's Law**: It quantitatively describes the electric force between two charges. The force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.
• **Electric Field**: It represents the region around a charged object where a force is exerted on other charged objects.
• **Electric Potential**: Known as voltage, it quantifies the work done to bring a unit positive charge from infinity to a particular point in space. It is crucial in calculating the potential energy between charges.

Understanding electrostatics is essential for solving problems relating to electric fields and potentials, as in the case of a charged metal ball.

Electric Field

The electric field is a vector quantity that provides a way to describe the influence of an electric charge on its surroundings. Think of it as a 'force field' that a charged object creates around itself, affecting other charges placed within it.

The strength and direction of an electric field are depicted using electric field lines. Electric fields originate from positive charges and terminate at negative charges, showing the path a positive test charge would take, if free to move.

Some important aspects of electric fields are:

• **Magnitude**: The strength of an electric field at any point in space determines how a charge at that point will feel a force. The field's magnitude is given by the equation: \[ E = \frac{F}{q} \] where \( F \) is the force experienced by a small test charge \( q \).
• **Direction**: Denoted by the orientation of field lines. In diagrams, lines closer together indicate a stronger field, while lines further apart indicate a weaker field.
• **Electric Field of a Point Charge**: This field is radial and reduces with the square of the distance. Given by: \[ E = \frac{kQ}{r^2} \] where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( r \) is the distance from charge.

Understanding the electric field allows one to predict how the presence of charges will affect nearby space and is essential for calculating electric potentials.

Charge Distribution

Charge distribution refers to how electric charges are arranged in a given space or on an object. It significantly impacts both the electric field and potential generated by an object.

When dealing with charge distribution, two common types are considered:

• **Point Charges**: Idealized charges that act at a single point in space. They simplify calculations and are used to model real-life situations like small charged particles.
• **Continuous Charge Distribution**: This involves charges spread over a volume, surface, or line, such as a charged metal sphere.
  The sphere acts as if all its charge were concentrated at its center, simplifying calculations of electric potential and field.

For a solid metal ball, understanding charge distribution helps determine how electric potential and field behave over and within the ball. In the example given, the potential on the surface and outside the ball results from the entire charge acting as if located at the center, making calculations feasible with formulas for point charges. Charge distribution dictating electric parameters demonstrates the importance of spatial charge arrangement in electrostatic problems.