An electric field can induce an electric dipole in a neutral atom or molecule by pushing the positive and negative charges in opposite directions. The dipole moment of an induced dipole is directly proportional to the electric field. That is, p → = α E → \overrightarrow{p}=\alpha\overrightarrow{E} , where α is called the polarizability of the molecule. A bigger field stretches the molecule farther and causes a larger dipole moment. An ion with charge q is distance r from a molecule with polarizability α. Find an expression for the force E → \overrightarrow{E} ion on dipole .

Hey everyone. So this problem tells us that an ion with a positive charge Q is located at a distance D from a molecule with polar is ability alpha. The ion generates an electric field capital E that induces an electric dipole in the molecule resulting in the separation of positive and negative charges within the molecule. The magnitude of the induced dipole moment can be expressed as P or moment is equal to alpha multiplied by E. We are asked to derive an expression for the force F sub ion exerted by the ion on the induced dipole. The expressions should be in terms of QD alpha and other relevant constants. Our multiple choice answers are shown here. So the first thing we can do with this problem is draw out what is happening. So we have a positively charged ion, it's interacting with a molecule that has positive and negative charges that are now separated. So we can recall that the force of the dipole on the ion is given by the equation F is equal to Q multiplied by E and therefore because of equal and opposite forces. So this was the force of the dipole on ion and were asked for the force of the exerted by the ion on the dipole. So that would be force of ion on dipole. And we'll write that just as F sub ion and that will be the opposite of the dipole on the ion. So negative Q multiplied by E. And so to solve for this expression, we need to solve for Q and E in terms of the other constants and variables in this problem. So we are told that the moment can be given as the equation P is equal to alpha multiplied by E. And we can recall that the energy at the dipole is equal to K multiplied by Q divided by D squared. And so if we combine these two equations, we have a moment which is equal to alpha kq all divided by D squared. And then the last equation that we need to recall to solve for this function for this expression is the electric field or E due to the dipole at a point on the axis of the dipole, which is given by the equation. So we'll call this the tai po two KP divided by D cubed. And so if we combine this equation for the electric field due to the dipole with this expression for the moment or sorry, the magnitude of the moment we come up with E is equal to two K multiplied by alpha K multiplied by Q divided by D cubed multiplied by D squared. And so we can simplify that and we get two K squared multiplied by alpha Q all divided by D to the fifth. And so going back to our original format, so F ion is equal to negative QE, we'll have negative Q multiplied by two K squared alpha Q divided by D to the fifth, which of course can simplify further to negative Q square, sorry, negative K squared multiplied by sorry negative two multiplied by K squared, multiplied by Q squared multiplied by alpha all divided by D to the fifth. And so that is the final expression for this scenario and it is negative and so it will be acting towards the ion. And so when we look at our multiple choice answers, we can see that, that aligns with answer choice B so B is the correct answer for this problem. That's all we have for this one. We'll see you in the next video.

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24. Electric Force & Field; Gauss' Law

Dipole Moment

Physics

24. Electric Force & Field; Gauss' Law

Dipole Moment

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Problem 61b

Textbook Question

An electric field can induce an electric dipole in a neutral atom or molecule by pushing the positive and negative charges in opposite directions. The dipole moment of an induced dipole is directly proportional to the electric field. That is, $\vec{p} = α \vec{E}$ , where α is called the polarizability of the molecule. A bigger field stretches the molecule farther and causes a larger dipole moment. An ion with charge q is distance r from a molecule with polarizability α. Find an expression for the force _{$\vec{E}$ ion on dipole}.

Verified step by step guidance

Step 1: Begin by understanding the relationship between the electric field created by the ion and the dipole moment induced in the molecule. The electric field due to a point charge q at a distance r is given by the formula:

E_{ion} = \frac{k q}{r^{2}}

, where k is Coulomb's constant.

Step 2: Use the relationship between the dipole moment and the electric field. The dipole moment induced in the molecule is given by:

p = α E_{ion}

. Substitute the expression for

E_{ion}

from Step 1 into this equation.

Step 3: Recognize that the force on a dipole in an electric field is given by the gradient of the potential energy. The potential energy of a dipole in an electric field is:

U = - p E_{ion}

. The force is then:

F = - \frac{d U}{dr}

Step 4: Substitute the expression for

U

into the force equation. This involves differentiating the product of

p

and

E_{ion}

with respect to r. Remember that

p

itself depends on

E_{ion}

, which is a function of r.

Step 5: Simplify the resulting expression to find the force. After substituting and differentiating, the force on the dipole due to the ion will be expressed in terms of q, r, α, and k. Ensure the final expression is consistent with the physical principles of electrostatics and dipole interactions.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Field

An electric field is a region around a charged particle where other charged particles experience a force. It is represented by the symbol E and is measured in volts per meter (V/m). The strength and direction of the electric field determine how charged particles, such as ions and dipoles, interact with each other. Understanding electric fields is crucial for analyzing how they induce dipoles in neutral atoms or molecules.

Dipole Moment

The dipole moment is a vector quantity that represents the separation of positive and negative charges within a system. It is defined as the product of the charge and the distance between the charges, denoted as p = qd. In the context of induced dipoles, the dipole moment is proportional to the strength of the electric field, as expressed by the equation p = αE, where α is the polarizability. This concept is essential for understanding how external electric fields influence molecular behavior.

Polarizability

Polarizability is a measure of how easily the electron cloud of a molecule can be distorted by an external electric field, resulting in an induced dipole moment. It is denoted by the symbol α and varies among different molecules based on their structure and electron distribution. A higher polarizability indicates that a molecule can develop a larger dipole moment in response to an electric field, which is critical for calculating the force exerted on the dipole by nearby charged particles.

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