Consider the configuration of current-carrying wires shown below. Two semi-infinite wires carry equal currents I and...

 

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Consider the configuration of current-carrying wires shown below. Two semi-infinite wires carry equal currents I₁ and I2, each with magnitude Io as shown. These wires meet at Point A, where their currents combine and split to form currents 13 and 14, which run down two semi-circular ares of radius a. These currents combine again at Point B to form 15 = 210, which is carried away in a third semi-infinite wire along the -y axis. I₁-10 13 R a 2R I R R B 12=10 13 = 210 On the left-hand arc, the upper and lower quarter circles of wire each have resistance R. On the right-hand arc, the upper quarter circle has resistance 2R and the lower quarter circle has resistance R. Thus, this system can be viewed as the following equivalent circuit: I₁ = 10 13 R 20 RL B I₂ = 210 1₂ = 10 (a) What is the equivalent resistance RAB measured between Points A and B? (b) What is the potential difference between Points A and B, AV = VA - VB? (c) What are the currents I3 and I? For the remainder of the problem, we will calculate the magnetic field B at the coor- dinate origin. For reference, the Biot-Savart law is: dB = olds×+ 1-2 (1) I₁ = 10 ds: 13 -to ds3 R a A 2R ds2 I₂ = 10 Q" R R x -Yo B ds; I₁ = 210 (d) Find the increment of the magnetic field dB3 at the coordinate origin arising from a very short section of the arc ds3 of length ds on the left-hand semicircular arc as shown in the figure above. (e) Guided by your result from Part (d), find the magnetic field B3 produced at the coordinate origin by the left-hand semicircular are of wire and the magnetic field B₁ produced at the coordinate origin by the right-hand semicircular arc of wire. (f) Find the increment of the magnetic field dB₁ at the coordinate origin arising from a very short segment ds of length dr at the position = -20 on the straight wire segment which carries the incoming current I₁. (g) Find the increment of the magnetic field dỄ₂ at the coordinate origin arising from a very short segment ds2 of length dr at the position x = +20 on the straight wire segment which carries the incoming current I2. (h) Find the increment of the magnetic field dB at the coordinate origin arising from a very short segment dss of length dy at the position y = -y on the straight wire segment which carries the outgoing current I5. (i) Now find the net magnetic field Bwires produced at the coordinate origin by the three straight wire segments, and hence find the total magnetic field B at the coordinate origin. Hint: you should be able to do this part without computing any complicated integrals. Consider the configuration of current-carrying wires shown below. Two semi-infinite wires carry equal currents I₁ and I2, each with magnitude Io as shown. These wires meet at Point A, where their currents combine and split to form currents 13 and 14, which run down two semi-circular ares of radius a. These currents combine again at Point B to form 15 = 210, which is carried away in a third semi-infinite wire along the -y axis. I₁-10 13 R a 2R I R R B 12=10 13 = 210 On the left-hand arc, the upper and lower quarter circles of wire each have resistance R. On the right-hand arc, the upper quarter circle has resistance 2R and the lower quarter circle has resistance R. Thus, this system can be viewed as the following equivalent circuit: I₁ = 10 13 R 20 RL B I₂ = 210 1₂ = 10 (a) What is the equivalent resistance RAB measured between Points A and B? (b) What is the potential difference between Points A and B, AV = VA - VB? (c) What are the currents I3 and I? For the remainder of the problem, we will calculate the magnetic field B at the coor- dinate origin. For reference, the Biot-Savart law is: dB = olds×+ 1-2 (1) I₁ = 10 ds: 13 -to ds3 R a A 2R ds2 I₂ = 10 Q" R R x -Yo B ds; I₁ = 210 (d) Find the increment of the magnetic field dB3 at the coordinate origin arising from a very short section of the arc ds3 of length ds on the left-hand semicircular arc as shown in the figure above. (e) Guided by your result from Part (d), find the magnetic field B3 produced at the coordinate origin by the left-hand semicircular are of wire and the magnetic field B₁ produced at the coordinate origin by the right-hand semicircular arc of wire. (f) Find the increment of the magnetic field dB₁ at the coordinate origin arising from a very short segment ds of length dr at the position = -20 on the straight wire segment which carries the incoming current I₁. (g) Find the increment of the magnetic field dỄ₂ at the coordinate origin arising from a very short segment ds2 of length dr at the position x = +20 on the straight wire segment which carries the incoming current I2. (h) Find the increment of the magnetic field dB at the coordinate origin arising from a very short segment dss of length dy at the position y = -y on the straight wire segment which carries the outgoing current I5. (i) Now find the net magnetic field Bwires produced at the coordinate origin by the three straight wire segments, and hence find the total magnetic field B at the coordinate origin. Hint: you should be able to do this part without computing any complicated integrals.