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1/27/17 Electric Fields (3) A. B. Kaye, Ph.D. Associate Professor of Physics 31January 2017 Tentative Schedule • Yesterday • • Coulomb’s Law First homework assigned • Today • Conclude Electric Fields • Tomorrow • • Gauss’ Law First homework is due ELECTRIC FIELDS Electric Field of a Continuous Charge Distribution 1 1/27/17 Electric Field – Continuous Charge Distribution • The distances between charges in a group of charges may be much smaller than the distance between the group and a point of interest. • In this situation, the system of charges can be modeled as continuous. • The system of closely spaced charges is equivalent to a total charge that is continuously distributed along some line, over some surface, or throughout some volume. Electric Field – Continuous Charge Distribution, cont How do we do this? Electric Field – Continuous Charge Distribution, equations • For the individual charge elements, we see that • … but because the charge distribution is continuous: 2 1/27/17 Charge Densities – Some Definitions • Linear charge density: when a charge is distributed along a line • l ≡ Q / ℓ with units of C/m • Surface charge density: when a charge is distributed evenly over a surface area • s ≡ Q / A with units of C/m2 • Volume charge density: when a charge is distributed evenly throughout a volume • r ≡ Q / V with units of C/m3 Amount of Charge in a Small Volume • If the charge is non-uniformly distributed over a line, surface, or volume, the amount of charge, dq, is given by: • • • For a length element: dq = l dℓ For a surface: dq = s dA For a volume: dq = r dV ELECTRIC FIELDS Electric Field Lines 3 1/27/17 Electric Field Lines • Field lines give us a means of representing the electric field pictorially • The electric field vector is tangent to the electric field line at each point • The line has a direction that is the same as that of the electric field vector • The number of lines per unit area through a surface perpendicular to the lines is proportional to the magnitude of the electric field in that region Electric Field Lines, General • The density of lines through surface A is greater than through surface B • The magnitude of the electric field is greater on surface A than B • The lines at different locations point in different directions • This indicates the field is non-uniform Electric Field Lines, Positive Point Charge • The field lines radiate outward in all directions • In three dimensions, the distribution is spherical • The lines are directed away from the source charge • A positive test charge would be repelled away from the positive source charge 4 1/27/17 Electric Field Lines, Negative Point Charge • The field lines are identical to that of a positive point charge, except for their direction • The field lines radiate inward (toward the point charge) in all directions • A positive test charge would be attracted toward the negative source charge Electric Field Lines – Rules for Drawing • The lines must begin on a positive charge (source) and terminate on a negative charge (drain) • In the case of an excess of one type of charge, some lines will begin or end infinitely far away • The number of lines drawn leaving a positive charge or approaching a negative charge is proportional to the magnitude of the charge • No two field lines can cross • Remember field lines are not material objects, they are a pictorial representation used to qualitatively describe the electric field Examples • Rules are great. How do we do draw these field lines in practice? • Let’s look at three examples: 1. A dipole arrangement of two charges with opposite signs 2. A pair of equal charges with the same sign 3. An unequal pair of charges with opposite signs 5 1/27/17 Electric Field Lines – Dipole • The charges are equal and opposite. • The number of field lines leaving the positive charge equals the number of lines terminating on the negative charge. Electric Field Lines – Like Charges • The charges are equal and positive • The same number of lines leave each charge • They are equal in magnitude • At a great distance, the field is approximately equal to that of a single charge of 2q • Since there are no negative charges available, the field lines end infinitely far away Electric Field Lines, Unequal Charges • The positive charge is twice the magnitude of the negative charge • Two lines leave the positive charge for each line that terminates on the negative charge • At a great distance, the field would be approximately the same as that due to a single charge of +q 6 1/27/17 ELECTRIC FIELDS Interesting Problems to Solve Interesting Problem #4 • A total amount of charge Q is distributed uniformly along the circumference of a thin glass ring of radius R. What is the electric field on the axis of the ring? Interesting Problem #5 • Positive charge is uniformly distributed over an infinite flat horizontal sheet, such as a very large sheet of paper. Suppose that the amount of charge per unit area on this sheet is σ. • Find the electric field in the space above and below the sheet. 7 1/27/17 ELECTRIC FIELDS Motion of a Charged Particle in a Uniform Electric Field Motion of Charged Particles • When a charged particle is placed in an electric field, it experiences an electrical force • If this is the only force on the particle, it must be the net force • The net force will cause the particle to accelerate according to Newton’s second law: • The acceleration is therefore: Motion of Particles, cont If the field is uniform, then the acceleration is constant • • A particle under constant acceleration model (from Physics I) can be applied to the motion of the particle • The electric force causes a particle to move according to the models of forces and motion • If the particle has a positive charge, its acceleration is in the direction of the field • If the particle has a negative charge, its acceleration is in the direction opposite the electric field 8 1/27/17 Electron in a Uniform Field, Example • An electron enters the region of a uniform electric field of strength E = 200 N/C with a velocity vi = 3.00 x 106 m/s. The plates (blue and red) have identical lengths of 0.100 m. 1. Find the acceleration of the electron while it is in the electric field. 2. Assuming the electron enters the field at a time t = 0, find the time at which it leaves the field. 3. Assuming the vertical position of the electron when it enters is yi = 0, what is the vertical position when it leaves the field? ELECTRIC FIELDS Interesting Problems to Solve Interesting Problem #6 • An ion milling machine uses a beam of gallium ions (m = 70 u) to carve microstructures from a target. A region of uniform electric field between parallel sheets of charge is used for precise control of the beam direction. Singly ionized gallium atoms with an initially horizontal velocity of 1.8 × 104 m/s enter a 2.0-cmlong region of uniform electric field which points vertically upward. The ions are redirected by the field and exit the field region at the angle θ. • If the field is set to a value of E = 90 N/C, what is the exit angle θ? 9 1/27/17 ELECTRIC FIELDS Key Review Points Key Review Points (1) • The electric field at some point in space is defined as the electric force that acts on a small, positive test charge at that point divided by the magnitude q0 of the test charge: • Coulomb’s Law describes the force interacting between static (i.e., non-moving) electrically charged particles: • The electric field due to a single point charge is the direction of the field is outward (inward) for positive (negative) charges • The potential energy of a dipole can be written as Key Review Points (2) • The electric field of an infinite, uniformly charged thin rod is the direction is perpendicular to the rod and outward (inward) for positive (negative) l • The electric field of an infinite, uniformly charged flat sheet is the direction is perpendicular to the sheet and outward (inward) for positive (negative) s • The electric field from a group of point charges can be computed by using superposition; i.e., the total net field at some point is the vector sum of the electric fields from all of the charges: 10 1/27/17 Key Review Points (3) • The electric field of a continuous charge distribution has a magnitude and the total field contributions to each component (e.g., can be computed by summing the ) with and where q is the angle between the electric field contribution and the x axis. • A linear charge distribution is given by dq = l dL, where l is in C/m • A surface charge distribution is given by dq = s dA, where s is in C/m2 • The electric dipole moment is from negative to positive • Torque on a dipole can be written as ; the direction of the dipole moment vector is Key Review Points (4) • The electric field associated with a pair of oppositely charged, parallel flat sheets is: • Motion in a uniform electric field is governed by • The Coulomb constant ke has a value of 8.9876 x 109 N・m2/C2 and can be written where e0 is the permittivity of free space with a value 8.8542 x 10–12 C2/N・m2 • The value of the fundamental free charge is e = 1.60218 x 10–19 C 11