ELECTRICITY & MAGNETISM (Fall 2011) LECTURE # 4 BY MOEEN GHIYAS

TODAY’S LESSON (Electric Field & Gauss Law) Fundamentals of Physics by Halliday / Resnick / Walker (Ch 23 / 24)

Today’s Lesson Contents Electric Field lines Motion of Charged Particles in Uniform Electric Field Electric Flux

Quiz # 1 Time :15 min

Electric Field Lines A convenient way of visualizing electric field patterns is to draw lines that follow the same direction as the electric field vector at any point. These lines, called electric field lines, are related to the electric field in any region of space in the following manner: –The electric field vector E is tangent to the electric field line at each point. –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. Thus, E is great when the field lines are close together and small when they are far apart.

Electric Field Lines Figure – Non uniform Electric Field –Electric field is more intense on surface A than on surface B –The lines at different locations point in different directions indicates that the field is non-uniform.

Electric Field Lines Figure - Electric field lines for the field due to a single charge –The lines are actually directed radially from the charge in all directions; thus, instead of the flat “wheel” of lines shown in 2 dimension, you should picture an entire sphere of lines

Electric Field Lines Figure - Electric field lines for the field due to a single charge –Why lines are directed outward for +ve charge and inward for -ve? –Because a positive test charge placed in this field would be repelled radially away from the positive point charge while same would be attracted toward the negative charge

Electric Field Lines Figure - Electric field lines for the field due to a single charge –In either case of +ve or –ve charge, the lines are along the radial direction and extend all the way to infinity. –Note that the lines become closer together as they approach the charge; this indicates that the strength of the field increases as we move toward the source charge.

Electric Field Lines The rules for drawing electric field lines are: –The lines must begin on a positive charge and terminate on a negative charge. –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.

Electric Field Lines Is this visualization of the electric field in terms of field lines consistent with Coulomb’s law? To answer this question, consider an imaginary spherical surface of radius r concentric with a point charge. From symmetry, we see that the magnitude of the electric field is the same everywhere on the surface of the sphere. The number of lines N that emerge from the charge is equal to the number that penetrate the spherical surface.

Electric Field Lines The number of lines N that emerge from the charge is equal to the number that penetrate the spherical surface. Hence, the number of lines per unit area on the sphere is N/4πr 2 (where the surface area of the sphere is 4πr 2 ). Because 4π is constant, E is proportional to the number of lines per unit area, i.e., E varies as 1/r 2 ; this finding is consistent with Coulomb’s law equation. Also for two charges, if object 1 has charge Q 1 and object 2 has charge Q 2, then ratio of number of lines is N 2 / N 1 = Q 2 / Q 1

Electric Field Lines Problems with model of qualitatively describing the electric field with the lines? 1.We draw a finite number of lines from (or to) each charge, which appear as if the field acts only in certain directions at certain places; instead, the field is continuous—that is, it exists at every point. 2.Danger of gaining the wrong impression from a two- dimensional drawing of field lines being used to describe a three-dimensional situation.

Electric Field Lines The lines are nearly radial at points close to either charge, The same number of lines emerge from each charge because the charges are equal in magnitude. At great distances from the charges, the field is approximately equal to that of a single point charge of magnitude 2q (q+q = 2q).

Electric Field Lines Now we sketch the electric field lines associated with a positive charge +2q and a negative charge -q. In this case, the number of lines leaving +2q is twice the number terminating at -q. Hence, only half of the lines that leave the positive charge reach the negative charge. The remaining half terminate on a negative charge we assume to be at infinity. At distances that are much greater than the charge separation, the electric field lines are equivalent to those of a single charge q (i.e. 2q-q = q).

Motion of Charged Particles in Uniform Electric Field When a particle of charge q and mass m is placed in an electric field E, We know E = F / q Then electric force on the charge is, F = E q But, from Newton’s 2 nd law we know F = m a Thus, we have F e = E q = m a The acceleration of the particle is thena = E q / m If E is uniform (i.e. constant in magnitude and direction), then the acceleration ‘a’ is constant. If q is +ve, then its acceleration is in the direction of the electric field, and opposite for negatively charged particle

Motion of Charged Particles in Uniform Electric Field The electric field in the region between two oppositely charged flat metallic plates is approximately uniform Suppose an electron of charge -e is projected horizontally into this field with an initial velocity v i i.

Motion of Charged Particles in Uniform Electric Field Because the electric field E in figure is in the +y direction, the acceleration of the electron is in the -y direction i.e. Since acceleration is constant, we can apply the equations of kinematics in two dimensions with v xi = v i and v yi = 0. After the time t, the components of its velocity are

Motion of Charged Particles in Uniform Electric Field Its coordinates after time t are Substituting the value t = x / v i from eqn 1 into eqn 2, we see that y is proportional to x 2. Hence, the trajectory is a parabola. After the electron leaves the field, it continues to move in a straight line in the direction of v, obeying Newton’s first law, with a speed v > v i

Motion of Charged Particles in Uniform Electric Field Since we are dealing with atomic particles, note that we have neglected the gravitational force acting on the electron. For an electric field of 10 4 N/C, the ratio of the magnitude of the electric force eE to the magnitude of the gravitational force mg is of the order of for an electron and of the order of for a proton.

Motion of Charged Particles in Uniform Electric Field Example – An Accelerated Electron – An electron enters the region of a uniform electric field as shown in fig, with v i = 3.00 x 10 6 m/s and E = 200 N/C. The horizontal length of the plates is l = m. (a) Find the acceleration of the electron while it is in the electric field. Solution:e = 1.60 x C, m = 9.11 x kg

Motion of Charged Particles in Uniform Electric Field Example – An Accelerated Electron – An electron enters the region of a uniform electric field as shown in fig, with v i = 3.00 x 10 6 m/s and E = 200 N/C. The horizontal length of the plates is l = m. (b) Find the time it takes the electron to travel through the field. Solution:

Electric Flux – (Uniform Electric Field) Electric field lines -qualitative method Electric flux -quantitative way Consider an electric field that is uniform in both magnitude and direction as shown The total number of lines penetrating the surface is proportional to the product EA Product of the magnitude of the electric field E and surface area A perpendicular to the field is called the electric flux Φ E : Φ E = EA From SI units flux is measured as Nm 2 /C

Electric Flux Example – What is the electric flux through a sphere that has a radius of 1.00 m and carries a charge of μC at its centre? Solution: Area of sphere = 4πr 2 = 12.6 m 2 Note: The field points radially outward and is therefore everywhere perpendicular to the surface

Electric Flux What if the surface through which electric field passing is not perpendicular to the field

Electric Flux Flux through a surface of fixed area A has –A max value EA when the surface is perpendicular to the field (when the normal to the surface is parallel to the field (θ = 0 0 ); –the flux is zero when the surface is parallel to the field (when the normal to the surface is perpendicular to the field (θ = 90 0 );

Electric Flux – (Non Uniform Electric Field) If the electric field varies (non-uniform) over a given surface and generally it does. Then, our definition of flux is applied only over a small element of area. Consider a general surface divided up into a large number of small elements, each of area ΔA. The variation in the electric field over one element can be neglected if the element is sufficiently small.

Electric Flux For vector ΔA i magnitude is the area of the ith element of the surface and direction is perpendicular to the surface element, as shown. The electric flux ΔΦ E through this element is By summing the contributions of all elements, we obtain the total flux through the surface.

Electric Flux If we let the area of each element approach zero, then the general definition of electric flux is Above equation is a surface integral, which means it must be evaluated over the surface in question.

Electric Flux We are often interested in evaluating the flux through a closed surface, that divides space into an inside and an outside region Consider a closed surface as in figure

Electric Flux The net flux through the surface is proportional to the net number of lines leaving the surface, where the net number means the number leaving the surface minus the number entering the surface. If more lines are leaving than entering, the net flux is positive. If more lines are entering than leaving, the net flux is negative.

Electric Flux Using the symbol to represent an integral over a closed surface, we can write the net flux Φ E as where E n represents the component of the electric field normal to the surface

Electric Flux Example – Consider a uniform electric field E oriented in the x direction. Find the net electric flux through the surface of a cube of edges ℓ, oriented as shown.

Electric Flux Solution:The net flux is the sum of the fluxes through all faces of the cube. First, note that flux through four of the faces (,, and the unnumbered ones) is zero because E is perpendicular to dA on these faces. Net flux through faces and is34 12

Summary / Conclusion Electric Field Lines Motion of Charged Particles in Uniform Electric Field Electric Flux