Point Charge & Dipole in an Electric Field Learning Objectives

Department of Physics United States Naval Academy Lecture 04: Point Charge & Dipole In An Electric Field

Learning Objectives • For a charged particle placed in an external electric field, apply the relationship between the electric field E~ at that point, the particle’s charge q, and the electrostatic force F~ that acts on the particle, and identify the relative directions of the force and the field when the particle is positively charged and negatively charged.

• For an electric dipole in an external field, calculate the torque by evaluating a cross product of the dipole moment vector and the electric field vector, in magnitude-angle notation and unit-vector notation. Also calculate the work done by the torque as the dipole rotates in the electric field. and estimate the potential energy of the system.

Electrostatic interaction: If a particle with charge q is placed in an external electric ﬁeld E~ , an electrostatic force F~ acts on it according to: F~ = qE~

Note that E~ is the electric field that other charges have produced at the location of the particle. (The field is not the field set up by the particle itself; to distinguish the two fields, the field acting on the particle is often called the external field. A charged particle or object is not affected by its own electric field.)

Direction of Force relative to Electric Field: If charge q is positive, the force vector, F~ is in the same direction as the ﬁeld vector, E~ . On the other hand, if charge q is negative, the force vector, F~ is in the opposite direction (the minus sign in the equation reverses the force vector from the ﬁeld vector).

Moving Charge in an Electric Field: A charged particle in an electric feels a force that is independent of its velocity but the path taken depends on the orientation of the velocity vector with respect to the field direction. The trajectory is curved for a charge particle whose velocity vector ~v is perpendicular to the electric field vector E~ . That is, the particles path is bent and the particle accelerates. On the other hand, if the velocity ~v is parallel to the electric field vector E~ , the path (direction) is unchanged but the still accelerates. The acceleration is due the influence of the electrostatic force on the particle.

The Electric Field Due to a Charged Disk: When an electric dipole of dipole moment ~p = qd~ is placed in an electric ﬁeld E~ , the ﬁeld exerts a torque ~τ on the dipole: ~τ = ~p × E~ = p E sin θ (Vector or Cross Product)

Here, θ is the angle between the dipole moment vector ~p and the electric field vector E~ . The rotational motion results from the fact that the dipole always wants its dipole moment to align with the external field (since the + and - charges want to go opposite ways but they are joined together!). See figure to the right.

The rotational motion of the dipole does not occur for free - energy is involved. This energy is stored in the form of potential energy that can be recovered when the ﬁeld disappears. Thus, the dipole potential energy U associated with its orientation in an electric ﬁeld E~ is: U = −~p · E~ = p E cos θ (Scalar or Dot Product)

This potential energy is deﬁned to be zero when ~p is perpendicular to E~ ; it is least (U = −pE) when ~p is aligned with E~ and greatest (U = pE) when ~p is directed opposite E~ .

The potential energies associated with four orientations of an electric dipole in an electric ﬁeld are (1) −5U0, (2) −7U0,

(3) 3U0, and (4) 5U0, where U0 is positive. Rank the orientations according to (a) the angle between the electric dipole moment ~p and the electric ﬁeld E~ and (b) the magnitude of the torque on the electric dipole, greatest ﬁrst.

An electron is accelerated eastward at 1.80 × 109 m/s2 by an electric ﬁeld. Determine the ﬁeld (a) magnitude and (b) direction.

A uniform electric ﬁeld exists in a region between two oppositely charged plates. An electron is released from rest at the surface of the negatively charged plate and strikes the surface of the opposite plate, 2.0 cm away, in a time 1.5 × 10−8 s. (a) What is the speed of the electron as it strikes the second plate? (b) What is the magnitude of the electric ﬁeld E~ ?

An electric dipole consisting of charges of magnitude 1.50 nC separated by 6.20 µm is in an electric ﬁeld of strength 1100 N/C. What are (a) the magnitude of the electric dipole moment and (b) the diﬀerence between the potential energies for dipole orientations parallel and antiparallel to E~ ?

An electron enters into a region with a uniform upward electric ﬁeld of magnitude 5.80E5 N/C. The initial velocity of the ◦ electron is 2.70E6 m/s directed 34.0 above the horizontal.

(a) the electron’s velocity as it passes through maximum height.

(b) the time it takes the electron to reach maximum height.

Two charges +2e and −2e are separated by a ﬁxed distance of 4.40 × 10−10 m forming an electric dipole. This dipole is ~ ˆ ◦ placed in a uniform electric ﬁeld E = (+5.80E5 N/C) i with the electric dipole moment ~p oriented at 34.0 above the +x axis.

(a) What is the energy of the dipole in this orientation?

(b) What is the magnitude of the torque on the dipole in this orientation? Describe the direction this torque would twist the dipole if the dipole were free to rotate.

(c) How much work would need to be done on the dipole to turn it to its maximum-energy orientation? Describe this orientation.

(d) If the dipole were allowed to freely rotate, what would be the dipoles rotational kinetic energy as it passes through its minimum-energy orientation? Describe this orientation.

(e) As it rotates through this minimum-energy orientation, what is the torque on the dipole at this instant?