Electromagnetism Lecture#3 Instructor: Engr. Muhammad Mateen Yaqoob

Mateen Yaqoob Department of Computer Science The Electric Field The concept of a field was developed by Michael Faraday (1791–1867) in the context of electric forces and is of such practical value. An electric field is said to exist in the region of space around a charged object, the source charge. When another charged object—the test charge—enters this electric field, an electric force acts on it Mateen Yaqoob Department of Computer Science

Mateen Yaqoob Department of Computer Science The Electric Field We define the electric field due to the source charge at the location of the test charge It has the SI units of newton per coulomb (N/C). The test charge serves as a detector of the electric field: an electric field exists at a point if a test charge at that point experiences an electric force. This equation gives us the force on a charged particle q placed in an electric field. If q is positive, the force is in the same direction as the field. If q is negative, the force and the field are in opposite directions. Mateen Yaqoob Department of Computer Science

Mateen Yaqoob Department of Computer Science The Electric Field We imagine using the test charge to determine the direction of the electric force and therefore that of the electric field. According to Coulomb’s law, the force exerted by q on the test charge is Mateen Yaqoob Department of Computer Science

Mateen Yaqoob Department of Computer Science Electric Field Lines Let’s now explore a means of visualizing the electric field in a pictorial representation. A convenient way of visualizing electric field patterns is to draw lines, called electric field lines and first introduced by Faraday, that are related to the electric field in a region of space in the following manner: Mateen Yaqoob Department of Computer Science

Mateen Yaqoob Department of Computer Science Electric Field Lines The lines are actually directed radially outward from the charge in all directions; therefore, instead of the flat “wheel” of lines shown, you should picture an entire spherical distribution of lines. Because a positive test charge placed in this field would be repelled by the positive source charge, the lines are directed radially away from the source charge. The electric field lines representing the field due to a single negative point charge are directed toward the charge. In either case, the lines are along the radial direction and extend all the way to infinity. Notice that the lines become closer together as they approach the charge, indicating that the strength of the field increases as we move toward the source charge. Mateen Yaqoob Department of Computer Science

Rules for drawing electric field lines The lines must begin on a positive charge and terminate on a negative charge. 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. Mateen Yaqoob Department of Computer Science

Mateen Yaqoob Department of Computer Science Electric Flux Concept of electric field lines was described qualitatively in previous lecture We now treat electric field lines in a more quantitative way Consider an electric field that is uniform in both magnitude and direction The field lines penetrate a rectangular surface of area A, whose plane is oriented perpendicular to the field Mateen Yaqoob Department of Computer Science

Mateen Yaqoob Department of Computer Science Electric Flux Number of lines per unit area (in other words, the line density) is proportional to the magnitude of the electric field. Therefore, the total number of lines penetrating the surface is proportional to the product EA This product of the magnitude of the electric field E and surface area A perpendicular to the field is called the electric flux Electric flux is proportional to the number of electric field lines penetrating some surface. SI Units of Electric Flux is newton-meters square per coulomb (N.m2/C) Mateen Yaqoob Department of Computer Science

Example: Electric Flux Through a Sphere What is the electric flux through a sphere that has a radius of 1.00 m and carries a charge of +1.00 µC at its center? Solution: The magnitude of the electric field 1.00 m from this charge is found using equation of electric field Mateen Yaqoob Department of Computer Science

Mateen Yaqoob Department of Computer Science Electric Flux The electric flux through this element is By summing the contributions of all elements, we obtain the total flux through the surface. If we let the area of each element approach zero, then the number of elements approaches infinity and the sum is replaced by an integral. Therefore, the general definition of electric flux is In general, value of electric flux depends both on field pattern and on surface Mateen Yaqoob Department of Computer Science

Example: Flux Through a Cube The net flux is sum of fluxes through all faces of cube. First, note that the flux through four of the faces Mateen Yaqoob Department of Computer Science

Mateen Yaqoob Department of Computer Science Gauss’s Law In today’s lecture we will describe a general relationship between the net electric flux through a closed surface (often called a gaussian surface) and charge enclosed by surface This relationship, known as Gauss’s law, is of fundamental importance in the study of electric fields Let us again consider a positive point charge located at center of a sphere of radius We know that magnitude of electric field everywhere on surface of sphere Mateen Yaqoob Department of Computer Science

Mateen Yaqoob Department of Computer Science Gauss’s Law Note from Equation 24.5 that the net flux through the spherical surface is propor- tional to the charge inside. The flux is independent of the radius r because the area of the spherical surface is proportional to r2 , whereas the electric field is proportional to 1/r2 . Thus, in the product of area and electric field, the dependence on r cancels. Mateen Yaqoob Department of Computer Science

Mateen Yaqoob Department of Computer Science Gauss’s Law Now consider several closed surfaces surrounding a charge Surface S1 is spherical, but surfaces S2 and S3 are not The flux that passes through S1 has the value q/Є0 As we discussed, flux is proportional to the number of electric field lines passing through a surface Therefore, we conclude that the net flux through any closed surface surrounding a point charge q is given by q/Є0 and is independent of the shape of that surface Mateen Yaqoob Department of Computer Science

Mateen Yaqoob Department of Computer Science Gauss’s Law Now consider a point charge located outside a closed surface of arbitrary shape As you can see from this construction, any electric field line that enters the surface leaves the surface at another point Number of electric field lines entering the surface equals number leaving surface Therefore, we conclude that the net electric flux through a closed surface that surrounds no charge is zero Mateen Yaqoob Department of Computer Science