Chapter 22: Electric Fields Physics 014 Chapter 22: Electric Fields

The Electric Field ? Q: How does particle 1 “know” of the presence of particle 2? That is, since the particles do not touch, how can particle 2 push on particle 1—how can there be such an action at a distance? Ans: particle 2 pushes on particle 1 not by touching it as you would push on a coffee mug by making contact. Instead, particle 2 pushes by means of the electric field it has set up.

Electric Field The electric field E at any point is defined in terms of the electrostatic force F that would be exerted on a positive test charge q0 placed there: Since E is the force per unit charge, it is measured in units of N/C.

Electric Field Lines The field lines of a positive charge radiate outward in all directions In three dimensions, the distribution is spherical The field lines of a negative charge radiate inward in all directions

Example 1. The isolated point charge of q=+15μC is in a vacuum Example 1. The isolated point charge of q=+15μC is in a vacuum. A test charge qo=+15μC is 0.20m to the right. Find the electric field at point p?

Electric Field Lines Field lines: useful way to visualize electric field E Field lines start at a positive charge, end at negative charge E at any point in space is tangential to field line Field lines are closer where E is stronger Example: a negative point charge — note spherical symmetry

Superposition Question: How do we figure out the field due to several point charges? Answer: consider one charge at a time, calculate the field (a vector!) produced by each charge, and then add all the vectors! (“superposition”) Useful to look out for SYMMETRY to simplify calculations!

Example 2 Total electric field -2q +q 4 charges are placed at the corners of a square as shown. What is the direction of the electric field at the center of the square? (a) Field is ZERO! (b) Along +y (c) Along +x -q +2q y x

Electric Field of a Dipole Electric dipole: two point charges +q and –q separated by a distance d Common arrangement in Nature: molecules, antennae, … Define “dipole moment” vector p: from –q to +q, with magnitude qd

Electric Field ON axis of dipole -q +q

Electric Field ON axis of dipole p = qa “dipole moment” -- VECTOR - + What if x>> a? (i.e. very far away) E~p/r3 is actually true for ANY point far from a dipole (not just on axis)

Continuous Charge Distribution Thus far, we have only dealt with discrete, point charges. Imagine instead that a charge Q is smeared out over a: LINE AREA VOLUME How to compute the electric field E?? Q Q Q Q

Electric Field – Continuous Charge Distribution, equations For a single charge For individual charge elements For continuous charge distribution

Charge Density Useful idea: charge density l = Q/L Useful idea: charge density Line of charge: charge per unit length = l Sheet of charge: charge per unit area = s Volume of charge: charge per unit volume = r s = Q/A r = Q/V

Charge Density Volume charge density: when a charge is distributed evenly throughout a volume ρ = Q / V with units C/m3 Surface charge density: when a charge is distributed evenly over a surface area σ = Q / A with units C/m2 Linear charge density: when a charge is distributed along a line λ = Q / ℓ with units C/m

Computing electric field of continuous charge distribution Approach: divide the continuous charge distribution into infinitesimally small elements Treat each element as a POINT charge & compute its electric field Sum (integrate) over all elements Always look for symmetry to simplify life!

Example: Field on Bisector of Charged Rod q L a P o y x dx Uniform line of charge +Q spread over length L What is the direction of the electric field at a point P on the perpendicular bisector? (a) Field is 0. (b) Along +y (c) Along +x Choose symmetrically located elements of length dx x components of E cancel

Example 3: Find the electric field due to a Line of Charge at point P? Uniform line of charge, length L, total charge Q Compute explicitly the magnitude of E at point P on perpendicular bisector Showed earlier that the net field at P is in the y direction -- let’s now compute this! y x a Q o L

- - - electric field due to a Line of Charge at P? Distance Q L a P o x dE dx d Charge per unit length

- - - electric field due to a Line of Charge at P? What is E very far away from the line (L<<a)? What is E if the line is infinitely long (L >> a)?

Problem-Solving Strategy Conceptualize Establish a mental representation of the problem Image the electric field produced by the charges or charge distribution Categorize Individual charge? Group of individual charges? Continuous distribution of charges?

Problem-Solving Strategy, cont Analyze Units: when using the Coulomb constant, ke, the charges must be in C and the distances in m Analyzing a group of individual charges: Use the superposition principle, find the fields due to the individual charges at the point of interest and then add them as vectors to find the resultant field Be careful with the manipulation of vector quantities Analyzing a continuous charge distribution: The vector sums for evaluating the total electric field at some point must be replaced with vector integrals Divide the charge distribution into infinitesimal pieces, calculate the vector sum by integrating over the entire charge distribution

Problem Solving Hints, final Analyze, cont. Symmetry: Take advantage of any symmetry to simplify calculations Finalize Check to see if the electric field expression is consistent with your mental representation Check to see if the solution reflects any symmetry present Image varying parameters to see if the mathematical result changes in a reasonable way

22-4 The Electric Field Due to a Line of Charge Example 4 – Find the electric field of a uniformly Charged Ring at point P? A differential element of charge occupies a length ds. This element sets up an electric field dE at point P. The components perpendicular to the z axis cancel. The parallel components add. © 2014 John Wiley & Sons, Inc. All rights reserved.

22-4 The Electric Field Due to a Line of Charge - - - electric field of a uniformly charged ring at P? Adding Components & replace cosθ . Integrating from a starting point (call it s=0) through the full circumference (s=2πR).

22-5 The Electric Field Due to a Charged Disk Example 5 – Find the electric field of a uniformly Charged Disc at point P? We superimpose a ring on the disk as shown The ring has radius r, radial width dr, and a charge element dq. It sets up a differential electric field dE at point P on its central axis. Then, we can sum all the dE contributions with We find © 2014 John Wiley & Sons, Inc. All rights reserved.

22-6 A Point Charge in an Electric Field If a particle with charge q is placed in an external electric field E, an electrostatic force F acts on the particle: Ink-jet printer. Drops shot from generator G receive a charge in charging unit C. An input signal from a computer controls the charge and thus the effect of field E on where the drop lands on the paper. © 2014 John Wiley & Sons, Inc. All rights reserved.

22-7 A Dipole in an Electric Field The torque on an electric dipole of dipole moment p when placed in an external electric field E is given by a cross product: A potential energy U is associated with the orientation of the dipole moment in the field, as given by a dot product: An electric dipole in a uniform external electric field E. Two centers of equal but opposite charge are separated by distance d. The line between them represents their rigid connection. Field E: causes a torque τ on the dipole. The direction of τ is into the page, as represented by the symbol (x-in a circle) . © 2014 John Wiley & Sons, Inc. All rights reserved.

© 2014 John Wiley & Sons, Inc. All rights reserved. Summary 22 Summary Definition of Electric Field The electric field at any point Field due to an Electric Dipole The magnitude of the electric field set up by the dipole at a distant point on the dipole axis is Eq. 22-1 Electric Field Lines provide a means for visualizing the directions and the magnitudes of electric fields Eq. 22-9 Field due to a Charged Disk The electric field magnitude at a point on the central axis through a uniformly charged disk is given by Field due to a Point Charge The magnitude of the electric field E set up by a point charge q at a distance r from the charge is Eq. 22-26 Eq. 22-3 © 2014 John Wiley & Sons, Inc. All rights reserved.

© 2014 John Wiley & Sons, Inc. All rights reserved. 22 Summary Summary Force on a Point Charge in an Electric Field When a point charge q is placed in an external electric field E Dipole in an Electric Field The electric field exerts a torque on a dipole The dipole has a potential energy U associated with its orientation in the field Eq. 22-34 Eq. 22-28 Eq. 22-38 © 2014 John Wiley & Sons, Inc. All rights reserved.

Adding vectors: Review Graphical method (triangle method): Draw vector A. Draw vector B starting at the tip of vector A. The resultant vector R = A + B is drawn from the tail of A to the tip of B.

Components of a vector The x- and y-components of a vector: The magnitude of a vector: The angle q between vector and x-axis:

Vector addition using unit vectors: Only add components!!!!! We want to calculate: R = A + B From diagram: R = (Ax + Bx)i + (Ay + By)j Rx = Ax + Bx Ry = Ay + By The components of R:

Vector addition using unit vectors: The magnitude of a R: The angle q between vector R and x-axis: