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Chapter 21: Problem 7

At one point in space, the electric potential energy of a \(15 \mathrm{nC}\) charge is \(45 \mu \mathrm{J}\) a. What is the electric potential at this point? b. If a \(25 \mathrm{nC}\) charge were placed at this point, what would its electric potential energy be?

Short Answer

Expert verified
a) The electric potential at this point is \(3000V\). b) The electric potential energy of a \(25 nC\) charge at this point would be \(75 \mu J\).

Step by step solution

01

Calculating the Electric Potential

The formula for electric potential (\(V\)) is given by \(V = U/Q\), where \(U\) is the electric potential energy and \(Q\) is the charge. Plugging in the given values: \(U = 45 \mu J = 45 \times 10^{-6} J\) and \(Q = 15 nC = 15 \times 10^{-9} C\), the electric potential then is \(V = (45 \times 10^{-6})/(15 \times 10^{-9}) = 3000 V\).
02

Calculating the Electric Potential Energy

Now, we can proceed to part b of the problem, which is to compute the electric potential energy of a \(25nC\) charge (\(Q'\)) placed at the point. We will use the reversed form of the previous formula: \(U' = V \times Q'\). Where \(V = 3000 V\) from Step 1 and \(Q' = 25 nC = 25 \times10^{-9} C\). Calculating \(U' = 3000 \times 25 \times 10^{-9}\), we get \(U' = 75 \mu J\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electric Potential
Electric potential is a measure of the potential energy per unit charge at a specific point in an electric field. Think of it like the electric 'pressure' that would act on a charged particle at that point鈥攊t's telling you how much potential energy a single unit of charge would have. It is denoted by the symbol 'V' and is expressed in units of volts (V). In physics, the formula to calculate electric potential is given by \( V = \frac{U}{Q} \), where \( U \) is the electric potential energy and \( Q \) is the electric charge.

In the provided exercise, the electric potential at a point where a 15 nC charge has an electric potential energy of 45 碌J is calculated. By using the values given and the formula, we find that the electric potential is 3000 V. This potential is intrinsic to the point in space and will exert the same 'pressure' on any other charge placed at this point.
Electric Charge
Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charges: positive and negative. Like charges repel each other, while opposite charges attract. The unit of charge is the Coulomb (C). Charges can also be expressed in terms of microcoulombs (碌C) or nanocoulombs (nC), which are just smaller subdivisions of the Coulomb.

In the context of our exercise, we work with charges in nanocoulombs. The initial charge in the problem is 15 nC, and in the second part of the problem, we consider a different charge of 25 nC. The relationship between charge and electric potential energy is critical because the energy depends not only on the electric potential but also on the magnitude of the charge placed in the potential.
Physics Problem Solving
Approaching physics problems often involves a step-by-step process that begins with understanding the concepts involved, identifying the relevant formulas, and then methodically applying these formulas to find the solution. It's essential to unpack the problem, organize the data, carry out the calculations carefully, and check the dimensions to ensure that the answer makes sense both numerically and conceptually.

In our example, the problem-solving steps are clearly outlined: first calculate the electric potential with the given energy and charge, and then use this electric potential to find the potential energy of a different charge at the same location. This structured approach not only yields the correct answers but also reinforces the underlying physical principles, allowing for a deeper understanding and the ability to solve similar problems in the future.

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Most popular questions from this chapter

Guiana dolphins are one of the few mammals able to detect electric fields. In a test of sensitivity, a dolphin was exposed to the variable electric field from a pair of charged electrodes. The magnitude of the electric field near the sensory organs was measured by detecting the potential difference between two measurement electrodes located \(1.0 \mathrm{cm}\) apart along the field lines. The dolphin could reliably detect a field that produced a potential difference of \(0.50 \mathrm{mV}\) between these two electrodes. What is the corresponding electric field strength?

A flying hummingbird picks up charge as it moves through the air. This creates a potential near the bird. What is the "voltage of a hummingbird"? Assume that the bird acquires a charge of \(+200 \mathrm{pC}\), a typical value, and model the bird as a sphere of radius \(3 \mathrm{cm}\).

Moving a charge from point A, where the potential is \(300 \mathrm{V}\), to point \(\mathrm{B},\) where the potential is \(150 \mathrm{V},\) takes \(4.5 \times 10^{-4} \mathrm{J}\) of work. What is the value of the charge?

The dielectric in a capacitor serves two purposes. It increases the capacitance, compared to an otherwise identical capacitor with an air gap, and it increases the maximum potential difference the capacitor can support. If the electric field in a material is sufficiently strong, the material will suddenly become able to conduct, creating a spark. The critical field strength, at which breakdown occurs, is \(3.0 \mathrm{MV} / \mathrm{m}\) for air, but \(60 \mathrm{MV} / \mathrm{m}\) for Teflon. a. A parallel-plate capacitor consists of two square plates, \(15 \mathrm{cm}\) on a side, spaced \(0.50 \mathrm{mm}\) apart with only air between them. What is the maximum energy that can be stored by the capacitor? b. What is the maximum energy that can be stored if the plates are separated by a 0.50-mm-thick Teflon sheet?

A 1.2 nF parallel-plate capacitor has an air gap between its plates. Its capacitance increases by \(3.0 \mathrm{nF}\) when the gap is filled by a dielectric. What is the dielectric constant of that dielectric?
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