Electric Fields

Gravitational Fields: Review Recall that surrounding any object with mass, or collection of objects with mass, is a gravitational field. Any mass placed in a gravitational field will experience a gravitational force. We defined the field strength as the gravitational force per unit mass on any “test mass” placed in the field: g = F / m. g is a vector that points in the direction of the net gravitational force; its units are N / kg. F is the vector force on the test mass, and m is the test mass, a scalar. g and F are always parallel. The strength of the field is independent of the test mass. For example, near Earth’s surface mg / m = g = 9.8 N / kg for any mass. Some fields are uniform (parallel, equally spaced fields lines). Nonuniform fields are stronger where the field lines are closer together. Earth Earth’s surface 10 kg 98 N m F uniform field nonuniform field

Electric Fields: Intro Surrounding any object with charge, or collection of objects with charge, is a electric field. Any charge placed in an electric field will experience a electrical force. We defined the field strength as the electric force per unit charge on any “test charge” placed in the field: E = F / q. E is a vector that points, by definition, in the direction of the net electric force on a positive charge; its units are N / C. F is the vector force on the test charge, and q is the test charge, a scalar. E and F are only parallel if the test charge is positive. Some fields are uniform (parallel, equally spaced fields lines) such as the field on the left formed by a sheet of negative charge. Nonuniform fields are stronger where the field lines are closer together, such as the field on the right produced by a sphere of negative charge q F q F uniform field nonuniform field + +

Overview of Fields Charge, like mass, is an intrinsic property of an object. Charges produce electric fields that affect other charges; masses produce gravitational fields that affect other masses. Gravitational fields lines always point toward an isolated mass. Unlike mass, though, charges can be positive or negative. Electric field lines emanate from positive charges and penetrate into negative charge. We refer to the charge producing a field as a field charge. A group of field charges can produce very nonuniform fields. To determine the strength of the field at a particular point, we place a small, positive test charge in the field. We then measure the electric force on it and divide by the test charge: For an isolated positive field charge, the field lines point away from the field charge (since the force on a positive charge would be away from the field charge). The opposite is true for an isolated negative field charge. No matter how complex the field, the electric force on a test charge is always tangent to the field line at that point. The coming slides will reiterate these ideas and provide examples. E = F / q.

Electric & Gravitational Fields Compared Field strength Force Intrinsic Property SI units g=W/mN / kg E=FEFE /qN / CN / C Gravity: Electric Force: Field strength is given by per unit mass or force per unit charge, depending on the type of field. Field strength means the magnitude of a field vector. Ex #1: If a +10 C charge is placed in an electric field and experiences a 50 N force, the field strength at the location of the charge is 5 N/C. The electric field vector is given by: E = 5 N/C, where the direction of this vector is parallel to the force vector (and the field lines). Ex #2: If a -10 C charge experiences a 50 N force, E = 5 N/C in a direction opposite the force vector (opposite the direction of the field lines).

Electric Field Example Problem A sphere of mass 1.3 grams is charged via friction, and in the process excess electrons are rubbed onto it, giving the sphere a charge of μC. The sphere is then placed into an external uniform electric field of 6 N/C directed to the right. The sphere is released from rest. What is its displacement after 15 s? (Hints on next slide.) E -

Sample Problem Hints q Eq E m gm g E - 1.Draw a vector as shown. Note that F E = q E, by definition of E, and that F E is to the left (opposite E ) since the charge is negative. 2.Instead of finding the net force (which would work), compute the acceleration due to each force separately. 3. Find the displacement due to each force using the time given and kinematics. 4. Add the displacement vectors to find the net displacement vector.

Drawing an E Field for a Point Charge + + Let’s use the idea of a test charge to produce the E field for an isolated positive field charge. We place small, positive test charges in the vicinity of the field and draw the force vector on each. Note that the closer the test charge is to the field charge, the greater the force, but all force vectors are directed radially outward from the field charge. At any point near the field charge, the force vector points in the direction of the electric field. Thus we have a field that looks like a sea urchin, with field lines radiating outward from the field charge to infinity in all direction, not just in a plane. The number of field lines drawn in arbitrary, but they should be evenly spaced around the field charge. What if the field charge were negative? Test charges and force vectors surrounding a field charge Isolated, positive point charge and its electric field

Single Positive Field Charge + This is a 2D picture of the field lines that surround a positive field charge that is either point-like or spherically symmetric. Not shown are field lines going out of and into the page. Keep in mind that the field lines radiate outwards because, by definition, an electric field vector points in the direction of the force on a positive test charge. The nearer you get to the charge, the more uniform and stronger the field. Farther away the field strength gets weaker, as indicated by the field lines becoming more spread out.

Single Negative Field Charge - The field surrounding an isolated, negative point (or spherically symmetric) charge looks just like that of an isolated positive charge except the field lines are directed toward the field charge. This is because, by definition, an electric field vector points in the direction of the force on a positive test charge, which, in this case is toward the field charge. As before, the field is stronger where the field lines are closer together, and the force vector on a test charge is parallel to the field.

Point Charges of Different Magnitudes Let’s compare the fields on two separate isolated point charges, one with a charge of +1 unit, the other with a charge of +2 units. It doesn’t matter how many field lines we draw emanating from the +1 charge so long as we draw twice as many line coming from the +2 charge. This means, at a given distance, the strength of the E field for the +2 charge is twice that for the +1 charge

Equal but Opposite Field Charges Pictured is the electric field produced by two equal but opposite charges. Because the charges are of the same magnitude, the field is symmetric. Note that all the lines that emanate from the positive charge land on the negative charge. Also pictured is a small positive charge placed in the field and the force vector on it at that position. This is the vector sum of the forces exerted on the test charge by each field charge. Note that the net force vector is tangent to the field line. This is always the case. In fact, the field is defined by the direction of net force vectors on test charges at various places. The net force on a negative test charge is tangent to the field as well, but it points in the opposite direction of the field. (Continued on next slide.) + - Link #1Link #1 Link #2Link #2Link #3

- + Equal but Opposite Field Charges (cont.) Here is another view of the field. Since the net force on a charge can only be in one direction, field lines never intersect. Draw the electric force on a positive charge at A, the electric field vector and B, and the electric force on a negative charge at C. The net force on a + charge at D charge is directly to the left. Show why this is the case by drawing force vectors from each field charge and then summing these vectors. A B C D

Two Identical Charges ++ With two identical field charges, the field is symmetric but all field lines go to infinity (if the charges are positive) or come from infinity (if the charges are negative). As with any field the net force on a test charge is tangent to the field. Here, each field charge repels a positive test charge. The forces are shown as well as the resultant vectors, which are tangent to the field lines.

Coulomb’s Law Review The force that two point charges, Q and q, separated by a distance r, exert on one another is given by: where K = 9  10 9 Nm 2 /C 2 (constant). F = K Q qK Q q r 2r 2 This formula only applies to point charges or spherically symmetric charges. Suppose that the force two point charges are exerting on one another is F. What is the force when one charge is tripled, the other is doubled, and the distance is cut in half ? Answer: 24 F

Field Strengths: Point Charge; Point Mass Suppose a test charge q is placed in the electric field produced by a point-like field charge Q. From the definition of electric field and Coulomb’s law K Q q / r 2K Q q / r 2 E = F q = q K QK Q r 2r 2 = Note that the field strength is independent of the charge placed in it. Suppose a test mass m is placed in the gravitational field produced by a point-like field mass M. From the definition of gravitational field and Newton’s law of universal gravitation G M m / r 2G M m / r 2 g = F m = m G MG M r 2r 2 = Again, the field strength is independent of the mass place in it.

Uniform Field Just as near Earth’s surface the gravitational field is approximately uniform, the electric field near the surface of a charged sphere is approximately uniform. A common way to produce a uniform E field is with a parallel plate capacitor: two flat, metal, parallel plates, one negative, one positive. Aside from some fringing on the edges, the field is nearly uniform inside. This means everywhere inside the capacitor the field has about the same magnitude and direction. Two positive test charges are depicted along with force vectors.

+ + More field lines emanate from the greater charge; none of the field lines cross and they all go to infinity. The field lines of the greater charge looks more like that of an isolated charge, since it dominates the smaller charge. If you “zoomed out” on this picture, i.e., if you looked at the field from a great distance, it would look like that of an isolated point charge due to one combined charge. Two + Field Charges of Different Magnitude Although in this pic the greater charge is depicted as physically bigger, this need not be the case.

Opposite Signs, Unequal Charges + - The positive charge has a greater magnitude than the negative charge. Explain why the field is as shown. (Answer on next slide.)

Opposite Signs, Unequal Charges (cont.) More field lines come from the positive charge than land on the negative. Those that don’t land on the negative charge go to infinity. As always, net force on a test charge is the vector sum of the two forces and it’s tangent to the field. Since the positive charge has greater magnitude, it dominates the negative charge, forcing the “turning points” of the point to be closer to the negative charge. If you were to “zoom out” (observe the field from a distance) it would look like that of an isolated, positive point with a charge equal to the net charge of the system. + -