Uniform Electric Fields The figure shows an electric field that is the same in strength and direction at every point in a region of space. This is called a uniform electric field. The easiest way to produce a uniform electric field is with a parallel-plate capacitor. Slide 26-74
Motion of a Charged Particle in an Electric Field Consider a particle of charge q and mass m at a point where an electric field E has been produced by other charges, the source charges. The electric field exerts a force F on q qe. Slide 26-79
Motion of a Charged Particle in an Electric Field The electric field exerts a force F on q qe on a charged particle. If this is the only force acting on q, it causes the charged particle to accelerate with In a uniform field, the acceleration is constant: Slide 26-80
Example 1 Two 2.0-cm-diameter disks face each other, 1.0 mm apart. They are charged to ±10 nc. A proton is shot from the negative disk toward the positive disk. What launch speed must the proton have to just barely reach the positive plate?
QuickCheck 26.11 A proton is moving to the right in a vertical electric field. A very short time later, the proton s velocity is Slide 26-82
QuickCheck 26.12 Which electric field is responsible for the proton s trajectory? A. B. C. D. Slide 26-84
Example 2 An electron is launched at a 45º angle with a speed of 5.0 x 10 6 m/s from the positive plate of the parallel-plate capacitor shown. The electron lands 4.0 cm away. What is the electric field strength inside the capacitor? What is the smallest possible spacing between the plates?
Dipoles in a Uniform Electric Field The figure shows an electric dipole placed in a uniform external electric field. The net force on the dipole is zero. The electric field exerts a torque on the dipole which causes it to rotate. Slide 26-86
Dipoles in a Uniform Electric Field The figure shows an electric dipole placed in a uniform external electric field. The torque causes the dipole to rotate until it is aligned with the electric field, as shown. Notice that the positive end of the dipole is in the direction in which E points. Slide 26-87
QuickCheck 26.13 Which dipole experiences no net force in the electric field? A. A. Dipole A. B. Dipole B. C. Dipole C. D. All three dipoles. B. C. Slide 26-88
QuickCheck 26.14 Which dipole experiences no net torque in the electric field? A. A. Dipole A. B. Dipole B. C. Dipole C. D. All three dipoles. B. C. Slide 26-90
Dipoles in a Uniform Electric Field The figure shows a sample of permanent dipoles, such as water molecules, in an external electric field. All the dipoles rotate until they are aligned with the electric field. This is the mechanism by which the sample becomes polarized. Slide 26-92
The Torque on a Dipole The torque on a dipole placed in a uniform external electric field is Slide 26-93
Dipoles in a Nonuniform Electric Field How will the dipole shown behave in the presence of the point charge shown? A. Rotate cw B. Rotate ccw C. Move to the left D. Move to the right E. More than one of the above. Slide 26-99
Example 3 Dipole Problem Three charges are placed at the corners of the triangle as shown in the figure below. Is the triangle in equilibrium? If so, explain why. If not, draw the equilibrium orientation. In equilibrium, will the triangle move to the right, to the left, or remain in place? Explain.
Electric Field of a Charged Cylinder Suppose we knew only two things about electric fields: 1. The field points away from positive charges, toward negative charges. 2. An electric field exerts a force on a charged particle. From this information alone, what can we deduce about the electric field of an infinitely long charged cylinder? All we know is that this charge is positive, and that it has cylindrical symmetry. Slide 27-19
Cylindrical Symmetry An infinitely long charged cylinder is symmetric with respect to: Translation parallel to the cylinder axis. Rotation by an angle about the cylinder axis. Reflections in any plane containing or perpendicular to the cylinder axis. The symmetry of the electric field must match the symmetry of the charge distribution. Slide 27-20
Electric Field of a Charged Cylinder Could the field look like the figure below? (Imagine this picture rotated about the axis.) The next slide shows what the field would look like reflected in a plane perpendicular to the axis (left to right). Slide 27-21
Electric Field of a Charged Cylinder This reflection, which does not make any change in the charge distribution itself, does change the electric field. Therefore, the electric field of a cylindrically symmetric charge distribution cannot have a component parallel to the cylinder axis. Slide 27-22
Electric Field of a Charged Cylinder Could the field look like the figure below? (Here we re looking down the axis of the cylinder.) The next slide shows what the field would look like reflected in a plane containing the axis (left to right). Slide 27-23
Electric Field of a Charged Cylinder This reflection, which does not make any change in the charge distribution itself, does change the electric field. Therefore, the electric field of a cylindrically symmetric charge distribution cannot have a component tangent to the circular cross section. Slide 27-24
Electric Field of a Charged Cylinder Based on symmetry arguments alone, an infinitely long charged cylinder must have a radial electric field, as shown below. This is the one electric field shape that matches the symmetry of the charge distribution. Side view End view Slide 27-25
Planar Symmetry There are three fundamental symmetries; the first is planar symmetry. Planar symmetry involves symmetry with respect to: Translation parallel to the plane. Rotation about any line perpendicular to the plane. Reflection in the plane. Slide 27-26
Cylindrical Symmetry There are three fundamental symmetries; the second is cylindrical symmetry. Cylindrical symmetry involves symmetry with respect to: Translation parallel to the axis. Rotation about the axis. Reflection in any plane containing or perpendicular to the axis. Slide 27-27
Spherical Symmetry There are three fundamental symmetries; the third is spherical symmetry. Spherical symmetry involves symmetry with respect to: Rotation about any axis which passes through the center point. Reflection in any plane containing the center point. Slide 27-28
Example 4 The figures show two cross sections of two infinitely long coaxial cylinders. The inner cylinder has a positive charge, the outer cylinder has an equal negative charge. Draw the correct electric field vectors using symmetry.
The Concept of Flux Consider a box surrounding a region of space. We can t see into the box, but we know there is an outward-pointing electric field passing through every surface. Since electric fields point away from positive charges, we can conclude that the box must contain net positive electric charge. Slide 27-29
The Concept of Flux Consider a box surrounding a region of space. We can t see into the box, but we know there is an inward-pointing electric field passing through every surface. Since electric fields point toward negative charges, we can conclude that the box must contain net negative electric charge. Slide 27-30
The Concept of Flux Consider a box surrounding a region of space. We can t see into the box, but we know that the electric field points into the box on the left, and an equal electric field points out of the box on the right. Since this external electric field is not altered by the contents of the box, the box must contain zero net electric charge. Slide 27-31
This box contains A. no net charge. B. a net negative charge. C. a net positive charge.
Example 5 What type of net charge does the box below possess? What electric field strength must be present through the front if a negative charge is inside? Which direction?