Question

In which situation is the electric potential the highest? a) at a point \(1 \mathrm{~m}\) from a point charge of \(1 \mathrm{C}\) b) at a point \(1 \mathrm{~m}\) from the center of a uniformly charged spherical shell of radius \(0.5 \mathrm{~m}\) with a total charge of \(1 \mathrm{C}\) c) at a point \(1 \mathrm{~m}\) from the center of a uniformly charged rod of length \(1 \mathrm{~m}\) and with a total charge of \(1 \mathrm{C}\) d) at a point \(2 \mathrm{~m}\) from a point charge of \(2 \mathrm{C}\) e) at a point \(0.5 \mathrm{~m}\) from a point charge of \(0.5 \mathrm{C}\)

Short answer

Answer: All situations have the same electric potential.

Step-by-step solution

Situation A: Point Charge of \(1 \mathrm{C}\)

In this case, we have a point charge of \(1\mathrm{C}\), and the distance from the charge is \(1\mathrm{~m}\). Using the formula for electric potential, \(V_A = \frac{k(1\mathrm{C})}{1\mathrm{~m}} = k\).

Situation B: Uniformly Charged Spherical Shell

In this situation, we have a uniformly charged spherical shell with a total charge of \(1\mathrm{C}\) and the distance from the center is \(1\mathrm{~m}\). Since the shell is uniformly charged, we can also use the formula for electric potential, considering the total charge \(Q\). Thus, \(V_B = \frac{k(1\mathrm{C})}{1\mathrm{~m}} = k\).

Situation C: Uniformly Charged Rod

For the uniformly charged rod, we can approximate the electric potential at the point \(1\mathrm{~m}\) from the center of the rod (as it is a uniformly charged rod with a length of \(1\mathrm{~m}\)) by using the point charge formula with total charge \(Q\). So, \(V_C= \frac{k(1\mathrm{C})}{1\mathrm{~m}} = k\).

Situation D: Point Charge of \(2 \mathrm{C}\)

In this situation, we have a point charge of \(2\mathrm{C}\), and the distance from the charge is \(2\mathrm{~m}\). Using the formula for electric potential, \(V_D =\frac{k(2\mathrm{C})}{2\mathrm{~m}} = k\).

Situation E: Point Charge of \(0.5 \mathrm{C}\)

In this case, we have a point charge of \(0.5\mathrm{C}\), and the distance from the charge is \(0.5\mathrm{~m}\). Using the formula for electric potential, \(V_E = \frac{k(0.5\mathrm{C})}{0.5\mathrm{~m}} = k\). Now, let's compare the electric potentials. We have: \(V_A = k\) ; \(V_B = k\) ; \(V_C = k\) ; \(V_D = k\) ; \(V_E = k\). All the electric potentials are equal to k. Therefore, there is no situation in which the electric potential is the highest, as they are all the same.

Key concepts

Point Charge

The concept of a **point charge** is fundamental in understanding electric fields and potentials. A point charge is an idealized model of a charged object that has negligible size. This hypothetical charge occupies no physical space but still creates an electric field around it, influencing other charges.
When calculating the electric potential due to a point charge, we use the formula \[ V = \frac{kQ}{r} \]where:

• \(V\) is the electric potential.
• \(k\) is Coulomb's constant, approximately \(8.99 \times 10^9 \mathrm{Nm}^2/\mathrm{C}^2\).
• \(Q\) is the point charge.
• \(r\) is the distance from the charge to the point in question.

Electric potential decreases as you move further from the charge. Thus, if you have a point charge of 1 C and are 1 meter away, the potential will be \( k \), as described in situation A of the problem. This concept is key to calculating potential in point charge situations in physics.

Spherical Shell

A **spherical shell** is another important concept in electrostatics, where a thin layer of charge is uniformly distributed over a spherical surface. This setup helps us understand how electric charge is distributed in spherical objects, like planets or metallic spheres.
For a spherical shell, the electric potential depends on where you are concerning the shell:

• Outside the shell: The shell behaves like a point charge located at the center, and the potential can be calculated as \( \frac{kQ}{r} \), similar to point charges.
• Inside the shell: The electric potential remains constant since the electric field inside a conductor is zero in electrostatic equilibrium.

In the problem's situation B, we consider a point 1 meter from the center of a charged spherical shell, which exhibits a potential of \( k \), as outside the shell acts like a point charge.

Charged Rod

A **charged rod** is a linear distribution of electric charge, which can be complex when calculating electric potential. For simplicity, we can approximate its interaction at a distance as similar to a point charge if the rod is uniform.
When you are calculating the electric potential at a point due to a charged rod, especially at a sizable distance compared to its length, you can simplify the problem using symmetry and integrate along the rod's length. However, if the rod is short relative to the distance to the point, using the point charge approximation is valid.
In this case, as described in situation C of the exercise, when the point is 1 meter away from a uniformly charged rod, we consider its potential equal to \( k \) when approximating the entire charge at one point. This approximation often simplifies the calculation while providing a generally accurate result in educational settings.

Electric Potential Formula

The **electric potential formula** is pivotal for solving numerous problems in electrostatics. Whether dealing with point charges, spherical shells, or rods, the formula remains fundamentally the same with some adjustments for different geometries:\[ V = \frac{kQ}{r} \]

• For a **point charge**, it's straightforward as discussed earlier.
• For a **spherical shell**, it's used for points outside the shell where the behavior mimics that of a point charge.
• For a **charged rod**, while integration may be needed for accuracy, the formula provides a baseline for potential approximations.

Finding the highest electric potential involves comparing these values at different distances and charge configurations, as seen in the exercise. Understanding this formula's application helps solve complex electrostatic problems by focusing on symmetry and distance.