Problem 40 - University Physics with Modern Physics

Question

Electric dipole moments of molecules are often measured in debyes \((\mathrm{D}),\) where \(1 \mathrm{D}=3.34 \cdot 10^{-30} \mathrm{C} \mathrm{m} .\) For instance, the dipole moment of hydrogen chloride gas molecules is \(1.05 \mathrm{D}\). Calculate the maximum torque such a molecule can experience in the presence of an electric field of magnitude \(160.0 \mathrm{~N} / \mathrm{C}\).

Step-by-Step Solution

Answer

Answer: The maximum torque experienced by a hydrogen chloride gas molecule in the presence of the given electric field is approximately \(5.60 \times 10^{-28} \mathrm{N} \cdot \mathrm{m}\).

Step 1: Convert dipole moment from debyes to Cm

First, let's convert the given dipole moment from debyes \((D)\) to Coulombs per meter \((Cm)\). Use the conversion factor \(1 \mathrm{D} = 3.34 \cdot 10^{-30} \mathrm{Cm}\): $$ \text{dipole moment in Cm} = 1.05\, \mathrm{D} \times \frac{3.34 \cdot 10^{-30} \mathrm{Cm}}{1\,\mathrm{D}} $$ Calculate the value: $$ \text{dipole moment in Cm} = 1.05 \times 3.34 \times 10^{-30} \mathrm{Cm} \approx 3.50 \times 10^{-30} \mathrm{Cm} $$

Step 2: Find the equation for torque on a dipole

The torque \(\tau\) exerted on an electric dipole in an electric field is given by: $$ \tau = p \cdot E \cdot \sin{\theta} $$ Where \(p\) is the dipole moment, \(E\) is the electric field, and \(\theta\) is the angle between the dipole moment and the electric field direction.

Step 3: Find the maximum torque

To find the maximum torque, we need to maximize the value of \(\sin{\theta}\). \(\sin{\theta}\) is maximized when \(\theta = 90^\circ\), in which case \(\sin{\theta} = 1\): $$ \tau_\text{max} = p \cdot E $$ Substitute the dipole moment (in Cm) and the electric field (in N/C) into the equation: $$ \tau_\text{max} = (3.50 \times 10^{-30} \mathrm{Cm}) \cdot (160.0 \mathrm{N/C}) $$ Calculate the maximum torque: $$ \tau_\text{max} \approx 5.60 \times 10^{-28} \mathrm{N} \cdot \mathrm{m} $$ So, the maximum torque experienced by a hydrogen chloride gas molecule in the presence of the given electric field is approximately \(5.60 \times 10^{-28} \mathrm{N} \cdot \mathrm{m}\).

Key Concepts

Torque on a Dipole

When discussing the forces acting on a dipole, understanding torque is crucial. Torque, represented by \(\tau\), is the measure of the rotational force acting on the electric dipole. It's like twisting a doorknob; here, the electric field acts like your hand applying force.
A dipole consists of two equal but opposite charges separated by a distance. In an electric field, these charges experience forces that try to rotate the dipole. This rotational effect is what we call torque. The formula used to calculate the torque on an electric dipole is:

• \(\tau = p \cdot E \cdot \sin{\theta}\)

Here, \(p\) is the dipole moment, \(E\) is the magnitude of the electric field, and \(\theta\) is the angle between \(p\) and \(E\).
Maximum torque occurs when the angle \(\theta\) is \(90^\circ\) since \(\sin{90^\circ} = 1\). In this situation, the dipole tends to align with the electric field. This concept is crucial when studying the behavior of polar molecules in electric fields.

Unit Conversion

Unit conversion is a vital skill in physics to ensure accuracy and consistency in calculations. It involves converting one unit of measurement into another, which can simplify or standardize values.
In this exercise, we began with converting the dipole moment from debyes \((\mathrm{D})\) to Coulombs-meters \((\mathrm{Cm})\). The conversion factor used is \(1 \mathrm{D} = 3.34 \cdot 10^{-30} \mathrm{Cm}\).
To convert:

• Multiply the given value in debyes by the conversion factor.
• Example: For a dipole moment of \(1.05 \mathrm{D}\), the conversion is \(1.05 \times 3.34 \times 10^{-30} \mathrm{Cm} = 3.50 \times 10^{-30} \mathrm{Cm}\).

Correct unit conversion is essential to ensure that all parts of a calculation “speak the same language,” ensuring that the results are valid and can be compared.

Electric Field

The electric field is a fundamental concept in electromagnetism that describes the influence a charged object exerts on other charges around it. It is a vector field, meaning it has both magnitude and direction.
The electric field \(E\) can be understood as the force per unit charge. It tells us how a positive test charge would move in the presence of another charge's field.
Key aspects of electric fields include:

• The strength of an electric field is measured in Newtons per Coulomb \((\mathrm{N/C})\).
• The direction of the field is defined as the direction a positive test charge would naturally move.
• Field lines visually represent the field; lines closer together indicate a stronger field.

In this scenario, the given electric field magnitude is \(160.0 \mathrm{N/C}\). This high-value aids in inducing a significant torque on molecules with a dipole moment as they strive to align with the field. Understanding electric fields helps in predicting how charges and dipoles interact under various conditions.