Fall 2008Lecture 1-1Physics 231 Electric Charges, Forces, and Fields

Fall 2008Lecture 1-2Physics 231 Electric Charges Electric charge is a basic property of matter Two basic charges Positive and Negative Each having an absolute value of 1.6 x Coulombs Experiments have shown that Like signed charges repel each other Unlike signed charges attract each other For an isolated system, the net charge of the system remains constant Charge Conservation

Fall 2008Lecture 1-3Physics 231 Two basics type of materials Conductors Materials, such as metals, that allow the free movement of charges Insulators Materials, such as rubber and glass, that don’t allow the free movement of charges

Fall 2008Lecture 1-4Physics 231 Coulomb’s Law Coulomb found that the electric force between two charged objects is Proportional to the product of the charges on the objects, and Inversely proportional to the separation of the objects squared k being a proportionality constant, having a value of x 10 9 Nm 2 /c 2

Fall 2008Lecture 1-5Physics 231 Electric Force This gives the force on charged object 2 due to charged object 1 The direction of the force is either parallel or antiparallel to this unit vector depending upon the relative signs of the charges is a unit vector pointing from object 1 to object 2 As with all forces, the electric force is a Vector So we rewrite Coulomb’s Law as q2q2 q1q1

Fall 2008Lecture 1-6Physics 231 Electric Force The force acting on each charged object has the same magnitude - but acting in opposite directions (Newton’s Third Law)

Fall 2008Lecture 1-7Physics 231 Example 1 A charged ball Q 1 is fixed to a horizontal surface as shown. When another massive charged ball Q 2 is brought near, it achieves an equilibrium position at a distance d 12 directly above Q 1. When Q 1 is replaced by a different charged ball Q 3, Q 2 achieves an equilibrium position at a distance d 23 (< d 12 ) directly above Q 3. For 1a and 1b which is the correct answer 1a: A) The charge of Q 3 has the same sign of the charge of Q 1 B) The charge of Q 3 has the opposite sign as the charge of Q 1 C) Cannot determine the relative signs of the charges of Q 3 & Q 1 1b: A) The magnitude of charge Q 3 < the magnitude of charge Q 1 B) The magnitude of charge Q 3 > the magnitude of charge Q 1 C) Cannot determine relative magnitudes of charges of Q 3 & Q 1 Q2Q2 Q1Q1 g d 12 Q2Q2 d 23 Q3Q3

Fall 2008Lecture 1-8Physics 231 To be in equilibrium, the total force on Q 2 must be zero. The only other known force acting on Q 2 is its weight. Therefore, in both cases, the electrical force on Q 2 must be directed upward to cancel its weight. Therefore, the sign of Q 3 must be the SAME as the sign of Q 1 A charged ball Q 1 is fixed to a horizontal surface as shown. When another massive charged ball Q 2 is brought near, it achieves an equilibrium position at a distance d 12 directly above Q 1. When Q 1 is replaced by a different charged ball Q 3, Q 2 achieves an equilibrium position at a distance d 23 (< d 12 ) directly above Q 3. 1a: A) The charge of Q 3 has the same sign of the charge of Q 1 B) The charge of Q 3 has the opposite sign as the charge of Q 1 C) Cannot determine the relative signs of the charges of Q 3 & Q 1 Q2Q2 Q1Q1 g d 12 Q2Q2 d 23 Q3Q3 Example 1

Fall 2008Lecture 1-9Physics 231 The electrical force on Q 2 must be the same in both cases … it just cancels the weight of Q 2 Since d 23 < d 12, the charge of Q 3 must be SMALLER than the charge of Q 1 so that the total electrical force can be the same!! A charged ball Q 1 is fixed to a horizontal surface as shown. When another massive charged ball Q 2 is brought near, it achieves an equilibrium position at a distance d 12 directly above Q 1. When Q 1 is replaced by a different charged ball Q 3, Q 2 achieves an equilibrium position at a distance d 23 (< d 12 ) directly above Q 3. 1b: A) The magnitude of charge Q 3 < the magnitude of charge Q 1 B) The magnitude of charge Q 3 > the magnitude of charge Q 1 C) Cannot determine relative magnitudes of charges of Q 3 & Q 1 Q2Q2 Q1Q1 g d 12 Q2Q2 d 23 Q3Q3 Example 1

Fall 2008Lecture 1-10Physics 231 More Than Two Charges q q1q1 q2q2 If q 1 were the only other charge, we would know the force on q due to q 1 - If q 2 were the only other charge, we would know the force on q due to q 2 - Given charges q, q 1, and q 2 What is the net force if both charges are present? The net force is given by the Superposition Principle

Fall 2008Lecture 1-11Physics 231 Superposition of Forces If there are more than two charged objects interacting with each other The net force on any one of the charged objects is The vector sum of the individual Coulomb forces on that charged object

Fall 2008Lecture 1-12Physics 231 Example Two q o, q 1, and q 2 are all point charges where q o = -1  C, q 1 = 3  C, and q 2 = 4  C What are F 0x and F 0y ? x ( cm ) y ( cm ) qoqo q2q2 q1q1  What is the force acting on q o ? We have that Decompose into its x and y components

Fall 2008Lecture 1-13Physics 231 Example Two - Continued Now add the components of and to find and X-direction: Y-direction: x ( cm ) y ( cm ) qoqo q2q2 q1q1

Fall 2008Lecture 1-14Physics 231 Example Two - Continued The magnitude of is Putting in the numbers... We then get for the components At an angle given by x ( cm ) y ( cm ) qoqo q2q2 q1q1

Fall 2008Lecture 1-15Physics 231 Note on constants k is in reality defined in terms of a more fundamental constant, known as the permittivity of free space.

Fall 2008Lecture 1-16Physics 231 Electric Field The Electric Force is like the Gravitational Force Action at a Distance The electric force can be thought of as being mediated by an electric field.

Fall 2008Lecture 1-17Physics 231 What is a Field? A Field is something that can be defined anywhere in space A field represents some physical quantity (e.g., temperature, wind speed, force) It can be a scalar field (e.g., Temperature field) It can be a vector field (e.g., Electric field) It can be a “tensor” field (e.g., Space-time curvature)

Fall 2008Lecture 1-18Physics A Scalar Field A scalar field is a map of a quantity that has only a magnitude, such as temperature

Fall 2008Lecture 1-19Physics A Vector Field A vector field is a map of a quantity that is a vector, a quantity having both magnitude and direction, such as wind

Fall 2008Lecture 1-20Physics 231 Electric Field We say that when a charged object is put at a point in space, The charged object sets up an Electric Field throughout the space surrounding the charged object It is this field that then exerts a force on another charged object

Fall 2008Lecture 1-21Physics 231 Electric Field Like the electric force, the electric field is also a vector If there is an electric force acting on an object having a charge q o, then the electric field at that point is given by (with the sign of q 0 included)

Fall 2008Lecture 1-22Physics 231 Electric Field The force on a positively charged object is in the same direction as the electric field at that point, While the force on a negative test charge is in the opposite direction as the electric field at the point

Fall 2008Lecture 1-23Physics 231 Electric Field A positive charge sets up an electric field pointing away from the charge A negative charge sets up an electric field pointing towards the charge

Fall 2008Lecture 1-24Physics 231 Electric Field The electric field of a point charge can then be shown to be given by Earlier we saw that the force on a charged object is given by The term in parentheses remains the same if we change the charge on the object at the point in question The quantity in the parentheses can be thought of as the electric field at the point where the test object is placed

Fall 2008Lecture 1-25Physics 231 Electric Field As with the electric force, if there are several charged objects, the net electric field at a given point is given by the vector sum of the individual electric fields

Fall 2008Lecture 1-26Physics 231 Electric Field If we have a continuous charge distribution the summation becomes an integral

Fall 2008Lecture 1-27Physics 231 Hints 1) Look for and exploit symmetries in the problem. 2) Choose variables for integration carefully. 3) Check limiting conditions for appropriate result

Fall 2008Lecture 1-28Physics 231 Electric Field Ring of Charge

Fall 2008Lecture 1-29Physics 231 Electric Field Line of Charge

Fall 2008Lecture 1-30Physics 231 Two equal, but opposite charges are placed on the x axis. The positive charge is placed at x = -5 m and the negative charge is placed at x = +5m as shown in the figure above. 1) What is the direction of the electric field at point A? a) up b) down c) left d) right e) zero 2) What is the direction of the electric field at point B? a) up b) down c) left d) right e) zero Example 3

Fall 2008Lecture 1-31Physics 231 Example 4 Two charges, Q 1 and Q 2, fixed along the x-axis as shown produce an electric field, E, at a point (x,y) = (0,d) which is directed along the negative y-axis. Which of the following is true? Q2Q2 Q1Q1 (c) E Q2Q2 Q1Q1 (b) E Q2Q2 Q1Q1 x y E d (a) Both charges Q 1 and Q 2 are positive (b) Both charges Q 1 and Q 2 are negative (c) The charges Q 1 and Q 2 have opposite signs E Q2Q2 Q1Q1 (a)

Fall 2008Lecture 1-32Physics 231 Electric Field Lines Possible to map out the electric field in a region of space An imaginary line that at any given point has its tangent being in the direction of the electric field at that point The spacing, density, of lines is related to the magnitude of the electric field at that point

Fall 2008Lecture 1-33Physics 231 Electric Field Lines At any given point, there can be only one field line The electric field has a unique direction at any given point Electric Field Lines Begin on Positive Charges End on Negative Charges

Fall 2008Lecture 1-34Physics 231 Electric Field Lines

Fall 2008Lecture 1-35Physics 231 Electric Dipole An electric dipole is a pair of point charges having equal magnitude but opposite sign that are separated by a distance d. Two questions concerning dipoles: 1) What are the forces and torques acting on a dipole when placed in an external electric field? 2) What does the electric field of a dipole look like?

Fall 2008Lecture 1-36Physics 231 Force on a Dipole Given a uniform external field Then since the charges are of equal magnitude, the force on each charge has the same value However the forces are in opposite directions! Therefore the net force on the dipole is F net = 0

Fall 2008Lecture 1-37Physics 231 Torque on a Dipole The individual forces acting on the dipole may not necessarily be acting along the same line. If this is the case, then there will be a torque acting on the dipole, causing the dipole to rotate.

Fall 2008Lecture 1-38Physics 231 Torque on a Dipole The torque is then given by  = qE dsin  d is a vector pointing from the negative charge to the positive charge

Fall 2008Lecture 1-39Physics 231 Potential Energy of a Dipole Given a dipole in an external field: Dipole will rotate due to torque Electric field will do work The work done is the negative of the change in potential energy of the dipole The potential energy can be shown to be

Fall 2008Lecture 1-40Physics 231 Electric Field of a Dipole