Unit 10: Static Electricity
Problem Set B
Overview:
Problem Set B targets your ability to use the perform complex analysis of physical situations by combining equations of electric force and electric field strength with trigonometry, vector principles and other physics principles learned throughout the course. The concepts and mathematics associated with electric force and electric field permeate the entire set. To be successful, you will have to understand these ideas.
A charged object can exert an attractive or repulsive force on other charged objects in its vicinity. The amount of force follows a rather predictable pattern which is dependent upon the amount of charge present on the two objects and the distance of separation. Coulomb's law of electric force expresses the relationship in the form of the following equation:
where F_{elect} represents the magnitude of the electric force (in Newtons), Q_{1} and Q_{2} represent the quantity of charge (in Coulombs) on objects 1 and 2, and d represents the separation distance between the objects' centers (in meters). The symbol k represents a constant of proportionality and has the value of 9.0 x 10^{9} N•m^{2}/C^{2}.
A charge object can exert an electric influence upon objects from which they are spatially separated. This action-at-a-distance phenomenon is sometimes explained by saying the charged object establishes an electric field in the space surrounding it. Other objects which enter the field interact with the field and experience the influence of the field. The strength of the electric field can be tested by measuring the force exerted on a test charge. Of course, the more charge on the test charge, the more force which would be experienced by it. While the force experienced by the test charge is proportional to the amount of charge on the test charge, the ratio of force to charge would be the same regardless of the amount of charge on the test charge. By definition, the electric field strength (E) at a given location about a source charge is simply the ratio of the force experienced (F) by a test charge to the quantity of charge on the test charge (q_{test}).
The electric field strength as created by a source charge (Q) varies with location. In accord with Coulomb's law, the force on a test charge is greatest when closest to the source charge and less when further away. Substitution of the expression for force into the above equation and subsequent algebraic simplification yields a second equation for electric field (E) which expresses its strength in terms of the variables which effect it. The equation is
where k is Coulombs constant of 9.0 x 10^{9} N•m^{2}/C^{2}, Q is the quantity of charge on the source creating the field and d is the distance from the center of the source.
Electric field and electric force are vector quantities which have a direction. In situations in which there are two or more force or field vectors present, it is often desired to know what the net electric force or field is. Finding the net value from knowledge of individual values requires that vectors be added together in head-to-tail fashion. If the vectors being added are at right angles to each other, then the Pythagorean theorem can be used to determine the resultant or net value; a trigonometric function can be used to determine an angle and subsequently a direction.
If the vectors being added are not at right angles to each other, then the usual procedure of adding them involves using a trigonometric function to resolve each vector into x- and y-components. The components are then added together to determine the sum of all x- and y-components. These sum values can then be added together in a right triangle to determine the net or resultant vector. And as usual, a trigonometric function can be used to determine an angle and subsequently a direction of the net or resultant vector. The graphic below depicts by means of diagrams how the components of a vector can be added together to determine the resultant of vectors A and B.
In addition to an understanding of the above mathematical quantities of charge, force and field, success on this problem set is dependent upon a conceptual understanding of the following principles.
Electric forces between objects can be attractive or repulsive. Objects charged with an opposite type of charge will be attracted to each other and objects charged with the same type of charge will be repelled by each other. These attractive and repulsive interactions describe the direction of the forces exerted upon any object.
Electric field is a vector quantity that has a directional nature associated with it. By convention, the direction of the electric field vector at any location surrounding a source charge is in the direction that a positive test charge would be pushed or pulled if placed at that location. Even if a negative charge is used to measure the strength of a source charge's field, the convention for direction is based upon the direction of force on a positive test charge.
Objects consisting of atoms containing protons and electrons can have an overall charge on them if there is an imbalance of protons and electrons. An object with more protons than electrons will be charged positively and an object with more electrons than protons will be charged negatively. The magnitude of the quantity of charge on an object is simply the difference between the number of protons and electrons multiplied by 1.6 x 10^{-19} C.
One convention commonly used for expressing the direction of vectors is the counterclockwise from east convention. The direction of a vector is represented as the counterclockwise angle of rotation which the vector makes with due East. The diagram below depicts several vectors and their approximate direction.
The force of gravity acting upon an object on Earth can be determined from its mass using the equation
where m is the mass of the object (in kg) and g is 9.8 m/s/s.
An object moving in a circle is experiencing a net force directed towards the center of the circle. The net force (in Newtons) can be related to the speed and radius of the circle by the equation
where m is the mass of the object (in kg), v is the speed of the object (in m/s) and R is the radius of the circle (in meters)
Additional Readings/Study Aids:
The following pages from The Physics Classroom tutorial may serve to be useful in understanding the mathematics of electric force and electric field.
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