Question

The amount of work done to move a positive point charge \(q\) on an equipotential surface of \(1000 \mathrm{~V}\) relative to that on an equipotential surface of \(10 \mathrm{~V}\) is a) the same. d) dependent on the b) less. distance the charge c) more. moves.

Short answer

Actually, the correct answer should be: d) dependent on the value of the charge q. The work done depends on the charge q, not on the distance the charge moves. I apologize for the mistake in the initial response.

Step-by-step solution

Identify the given variables and constants.

Here, we are given: Voltage of surface 1, V1 = 1000 V Voltage of surface 2, V2 = 10 V Charge: q

Calculate the potential difference between the two surfaces.

To find the potential difference between the two equipotential surfaces, subtract the potential of surface 2 from the potential of surface 1. ∆V = V1 - V2 = 1000 V - 10 V = 990 V

Calculate the work done to move the charge between the two surfaces.

Now we can calculate the work done (W) to move the positive point charge q between these two surfaces. W = q∆V Since this formula includes q, and no further information is given about q or the distance moved, the work done will depend on the charge.

Identify the answer in the options.

Based on the calculations, the amount of work done to move the positive point charge q between the two equipotential surfaces is dependent on the charge. Thus, the correct answer is: d) dependent on the distance the charge moves.

Key concepts

Electric Potential Difference

To understand the concept of electric potential difference, let's consider a familiar analogy. Imagine you have two water tanks at different heights. The difference in height creates a pressure difference, which could potentially push water from the high tank to the low one if a path is open. Similarly, in the world of electricity, the electric potential difference (often just called voltage) between two points is what 'pushes' electric charges from a point of high potential energy to low potential energy.

Using the formula for potential difference, \( \Delta V = V_1 - V_2 \) where \( V_1 \) and \( V_2 \) are the potentials at two different points, we find that moving a charge between these two points involves a specific change in potential. Going back to our textbook exercise, a positive point charge moving from a spot with 1000 V to one with 10 V experiences a potential difference of 990 V. This difference is what 'encourages' the charge to move, just like water naturally flows downhill.

Work Done by Electric Force

Imagine pushing a ball up a hill. The work you do is equal to the force you apply times the distance you push it. In the electrical world, we define work done by electric force similarly, using the equation \( W = q\Delta V \) where \( W \) is the work done by the electric force, \( q \) is the charge, and \( \Delta V \) is the electric potential difference.

When you're considering work done on a charge in an electric field, it's crucial to realize that if the electric potential doesn't change—as on an equipotential surface—no work is needed to move that charge on that surface. That's because the force the electric field exerts on the charge at every point on that surface does not have a component in the direction of the movement, just like you do no work balancing a book on your head while walking flat ground. But, as soon as there is a potential difference, as with our 990 V from earlier, work comes into play, and it is proportional to both the charge and the electric potential difference.

Positive Point Charge

Exploring the role of a positive point charge in this electrical dance, it helps to first clarify what we mean by a point charge. A point charge is a charged object considered to be dimensionless; it serves as an idealization to simplify the analysis of electric fields. A positive point charge, specifically, has an excess of protons and is drawn toward areas of lower electric potential, or towards negative charges.

In our exercise scenario, the positive point charge is the 'character' moving from one equipotential surface to another. The electric potential difference acts somewhat like a downhill slope for this charge. Since work is done on this charge as it moves between different voltages, and the amount of work depends on the magnitude of the charge itself, it's important to remember that charges in electric fields don't just float around aimlessly—they respond actively to the 'landscape' of electric potential.