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# Homework 2 - 1 A square with each side of length L has four...

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- 1 - 1. A square with each side of length L has four charges fixed in place at each corner. At two of the corners we have a negative charge – Q, while a third corner has a positive charge twice as large (+ 2Q). In the final corner there is a different charge q . (i) Find the magnitude of the net force on charge q . (ii) The total force on the +2Q charge is zero. What must q be for this to be true? 2. Two very small spheres, each of mass m , are suspended from a common point by insulating threads of length L . When each sphere carries a charge q , the threads each make an angle of q with the vertical. (Note: assume the radius of each sphere is negligible relative to L .) (i) By drawing a clear free-body diagram for one of the spheres, show that the string tension in either thread is θ = cos mg F T Drawing the free body diagram for the right sphere The y-equation for equilibrium is 0 mg - cos F T = θ Rearrange to get θ = cos mg F T (ii) Show that the magnitude of the charge q is given by k mg L q θ θ = tan sin 2 . The x-equation for equilibrium is 0 sin ) sin (2L q 2 2 = θ θ T F k Rearrange to get k mg L q θ θ = tan sin 2 – Q – Q +2 Q q L F 1 = 2 L Q q k (+ y ˆ ) F 2 = 2 L Q q k (+ x ˆ ) F 3 = 2 L) 2 ( (2Q) q k ( - 2 2 x ˆ - 2 2 y ˆ ) F NET = 2 L Q q k (1 - 2 2 ) ( x ˆ + y ˆ ) F NET 2Q = 2 2 L Q 2 k (- x ˆ - y ˆ ) + 2 2L 2Q q k ( + 2 2 x ˆ + 2 2 y ˆ ) just set the ( x ˆ ) part to zero (the ( y ˆ ) equation is identical) 2 2 L 2Q k = 2 2L Q q 2 k 2 2 q = Q 2 4 = Q 2 2 θ F T F E m g

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- 2 - 3. Two equal charges + Q are at two vertices of an equilateral triangle with sides a; a third charge – q is at the other vertex. A fourth charge q 0 , located at a /2 outside the triangle along the bisector of the + Q charges (as shown) experiences zero net force. Find the ratio Q q Assume the charge q 0 is positive.
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