Problem 72 - University Physics with Modern Physics

Question

Carbon monoxide (CO) has a dipole moment of approximately \(8.0 \cdot 10^{-30} \mathrm{C} \mathrm{m} .\) If the two atoms are separated by \(1.2 \cdot 10^{-10} \mathrm{~m}\), find the net charge on each atom and the maximum amount of torque the molecule would experience in an electric field of \(500.0 \mathrm{~N} / \mathrm{C}\).

Step-by-Step Solution

Answer

Answer: The net charge on each atom in the CO molecule is approximately \(6.67 \cdot 10^{-20} \, \mathrm{C}\), and the maximum torque it would experience in the given electric field is approximately \(4.0 \cdot 10^{-27} \, \mathrm{N}\, \mathrm{m}\).

Step 1: Calculate the net charge on each atom

We are given the dipole moment, \(p\), and the distance between atoms, \(r\), which can be used to find the net charge on each atom. The dipole moment is defined as the product of the net charge, \(q\), and the distance between the charges, \(r\): \(p = qr\) We can rearrange this equation to find the net charge: \(q = \dfrac{p}{r}\) Now, we'll plug in the given values for \(p\) and \(r\): \(p = 8.0 \cdot 10^{-30} \, \mathrm{C}\, \mathrm{m}\) \(r = 1.2 \cdot 10^{-10} \, \mathrm{m}\) \(q = \dfrac{8.0 \cdot 10^{-30}}{1.2 \cdot 10^{-10}}\) \(q = 6.67 \cdot 10^{-20} \, \mathrm{C}\) So, the net charge on each atom is approximately \(6.67 \cdot 10^{-20} \, \mathrm{C}\).

Step 2: Calculate the maximum torque experienced by the molecule in an electric field

The torque experienced by a dipole in an electric field, \(E\), is given by the formula: \(\tau = pE\sin{\theta}\) Where \(\theta\) is the angle between the dipole moment and the electric field. The maximum torque will be experienced when \(\theta = 90^\circ\), and sin(\(90^\circ\)) = 1. We are given the electric field, \(E = 500.0 \, \mathrm{N/C}\), and we have already calculated the dipole moment, \(p\). Thus, we can find the maximum torque: \(\tau_\text{max} = pE\sin{90^\circ}\) \(\tau_\text{max} = (8.0 \cdot 10^{-30} \, \mathrm{C}\, \mathrm{m})(500.0 \, \mathrm{N/C})(1)\) \(\tau_\text{max} = 4.0 \cdot 10^{-27} \, \mathrm{N}\, \mathrm{m}\) The maximum torque the CO molecule would experience in the given electric field is approximately \(4.0 \cdot 10^{-27} \, \mathrm{N}\, \mathrm{m}\).

Key Concepts

Electric Charge

Electric charge is one of the fundamental properties of matter, related to the electromagnetic force, one of the four fundamental forces of nature. It's quantified in coulombs (C) and comes in two types, commonly referred to as positive and negative charges. Charges with the same sign repel each other, whereas opposite charges attract.
Particles such as electrons and protons carry charge, with electrons having a negative charge and protons having a positive charge. In atoms, the number of protons and electrons are usually the same, balancing out to give a net charge of zero. In the case of the carbon monoxide molecule from our exercise, the net charge on each atom isn't perfectly balanced, leading to the formation of a dipole moment.
Understanding the concept of electric charge is crucial when working with electrical systems or exploring the behaviors of molecules under the influence of electric fields, as the charge determines how they would interact with the field.

Electric Field

An electric field is a region around a charged particle or object within which a force would be exerted on other charged particles or objects. The strength of an electric field is measured in newtons per coulomb (N/C) and represents the force per unit charge it would exert.
The direction of the electric field is defined as the direction a positive test charge would move if placed within the field. For our carbon monoxide molecule, the electric field would act on the dipole, exerting force on each charge, which could potentially cause the molecule to align with the field. This concept is pivotal for understanding how electric charges interact in the presence of other charges, and it applies widely, from explaining the structure of atoms to understanding the operation of electronic devices.

Torque on a Dipole

Speaking of torque in the context of a dipole in an electric field, it refers to the twisting force that tends to cause rotational motion. The dipole moment, electric field strength, and the angle between them determine the magnitude of the torque. The formula for torque on a dipole is
\[\begin{equation}\tau = pE\sin{\theta}\end{equation}\]where \(\tau\) is the torque, \(p\) is the dipole moment, \(E\) is the electric field strength, and \(\theta\) is the angle between the dipole moment vector and the electric field vector. The maximum torque occurs when this angle is 90 degrees, as sine of 90 degrees is equal to 1.
In practical applications, torque on a dipole is an essential concept for understanding the behavior of molecules in fields like molecular biology, chemistry, and materials science, where the orientation of molecules in external fields can be crucial.