Example 17-5 Electric Potential Difference in a Uniform Field II Determine the electric potential difference V = Vc 2 Va between points a and c in the uniform electric field of Example 17-4. Point c is a distance 0.500 m from point a, and the straight-line path from a to c makes an angle of 126.9° with respect to the electric field. Set Up Again we’ll use Equation 17-7 to calculate the potential difference between the two points. Since the displacement from point a to point c points generally opposite to the direction of the electric field, we expect that V = Vc 2 Va will be positive, just like V = Vb 2 Va in Example 17-4. Solve Use Equation 17-7 to solve for the potential difference. E = 2.00 × 102 V/m Potential difference between two points in a uniform electric field: V = Vc 2 Va = 2Ed cos u a (17-7) d = 0.500 m O = 126.9° c Calculate the potential difference from the electric field magnitude E, the displacement d, and the angle u: V = Vc 2 Va = 2Ed cos u = - 12.00 * 102 V>m2 10.500 m2 cos 126.9 = - 12.00 * 102 V>m2 10.500 m2 1 -0.6002 = +60.0 V Reflect The potential difference between points a and c is the same as the potential difference between points a and b in Example 17-4. Equation 17-7 tells us why this should be: The potential difference V = 2Ed cos u involves the magnitude s. In both E of the electric field multiplied by d cos u, which is the component of the displacement ds in the direction of E 2 examples the electric field magnitude has the same value (2.00 * 10 V>m), as does the component of displacement in the direction of the electric field (20.300 m). Another way to come to this same conclusion is to recognize that the displacement from a to c can be broken down into two parts: a displacement in the negative x direction from a to b, followed by a displacement from b to c in the y direction. The potential difference for the displacement from a to b is Vb 2 Va = +60.0 V as we calculated in Example 17-4; the potential difference for the displacement from b to c is Vc 2 Vb = 2Ed cos 90° = 0 since that displacement is perpendicular to the electric field. So the net potential difference between a and c is Vc 2 Va = (Vc 2 Vb) + (Vb 2 Va) = +60.0 V + 0 V = +60.0 V.