A point charge q is situated a large distance r from a neutral atom of polarizability α. Find th

A point charge q is situated a large distance r from a...

Key Concepts

Electric Field of a Point Charge

This concept refers to the electric field generated by a point charge, which in free space is given by Coulomb’s law. The field strength decreases with the square of the distance from the charge and acts radially outward (or inward for a negative charge). Understanding this field is fundamental to determining the influence a point charge exerts on its surroundings.

Induced Dipole Moment

A neutral atom or molecule can develop a dipole moment when exposed to an external electric field. The induced dipole moment is proportional to both the atomic or molecular polarizability and the applied electric field. This concept is key when considering how nonpolar atoms respond to nearby charges.

Atomic Polarizability

Atomic polarizability is a measure of how easily the electron cloud in an atom or molecule can be distorted by an external electric field. It quantifies the induced dipole moment per unit of electric field strength and plays a crucial role in determining the strength of induced dipole interactions.

Potential Energy of an Induced Dipole in an Electric Field

The energy associated with an induced dipole in an electric field is given by U = -½ ?E². The factor of ½ arises because the dipole moment is induced by the field and builds up as the field is applied. This potential energy is essential for analyzing the interaction between the induced dipole and the external field.

Force from the Energy Gradient

The force acting on a dipole in a non-uniform electric field is determined by the gradient of the potential energy, F = -dU/dr. By differentiating the energy with respect to distance, one obtains the attractive or repulsive force acting on the atom or molecule. This concept bridges the energy description of interactions and the resulting mechanical forces.

Transcript

00:01 One of the things that's true about a charged object held near a polarizable material is there's always a force of attraction.

00:13 And so we're going to develop a little model that explains this.

00:17 We're going to bring a point charge somewhere in the vicinity of a neutral molecule, which can be polarized.

00:28 And a reminder that polarizability relates the dipole moment of that molecule p to the external electric field in which that molecule is sitting through the polarizability alpha.

00:49 So first off, that dipole acquires a dipole moment because of the electric field.

01:00 Of the point charge.

01:09 And that dipole moment will be aligned such that the negative end of the dipole is closer to the positive point charge, just for the sake of argument.

01:23 But we'll kind of not worry too much about the actual details since we're making a general model.

01:32 But anyway, the point charge electric field is of course just the kulam relationship, q over 4 pi epsilon not, divided by the inverse squared of the separation.

01:52 So we now have an expression for the dipole moments.

02:06 Now what's going to create the attraction is the electric field of the dipole on the point charge.

02:15 So if we write down the force, on plus q, it's going to be equal to plus q times the electric field of the dipole.

02:33 And recalling a previous relationship for this electric field, we can write it down in terms of spherical coordinates or coordinate free representation, if we prefer.

02:52 But that field basically goes as the dipole moment over arc cubed, and then some vector relationships.

03:13 I started off drawing this picture with everything aligned along the z axis on purpose...

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