PHI 1101 : Logical Symbols

An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are q and −q and are located at a distance d from each other, then the electric field E at the point P in the figure is . 

By expanding this expression for E as a series in powers of d/D, show that E is approximately proportional to 1/D³ when P is far away from the dipole.

1. A student has mapped out a few magnetic field lines using a compass, as shown in the figure.

Rank points C, D, and E according to the magnitude of the magnetic field at those locations, from greatest to least (for example, if there were points P, Q and R, and the field at P and R were the same and both were greater than the field at Q, you would write "P = R > Q").

2. The student then places the compass at point C. Which of the following best represents the orientation of the compass needle?

3. The student then moves the compass to point D. Which of the following best represents the orientation of the compass needle?

4. The student then places the compass inside a hollowed-out magnet as shown.

 Which of the following best represents the orientation of the compass needle?

Pierre de Fermat (1601–1665) showed that whenever light travels from one point to another, its actual path is the path that requires the smallest time interval. This statement is known as Fermat’s principle. The simplest example is for light propagating in a homogeneous medium. It moves in a straight line because a straight line is the shortest distance between two points. Derive Snell’s law of refraction from Fermat’s principle. Proceed as follows. In Figure P35.74, a light ray travels from point P in medium 1 to point Q in medium 2. The two points are, respectively, at perpendicular distances a and b from the interface. The displacement from P to Q has the component d parallel to the interface, and we let x represent the coordinate of the point where the ray enters the second medium. Let t 5 0 be the instant the light starts from P.

(a) Show that the time at which the light arrives at Q is 

(b) To obtain the value of x for which t has its minimum value, differentiate x with respect to t and set the derivative equal to zero. Show that the result implies

(c) Show that this expression in turn gives Snell's law,