(1) A circuit has resistors ? 20.0 Ω and ? ? 10.0 Ω, a capacitor ? 10.0 ?F, and a battery of EMF ℇ 5.00 V as shown in the figure. Assume the capacitor is a parallel-plate capacitor, and it starts off...

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(1) A circuit has resistors ? 20.0 Ω and ? ? 10.0 Ω, a capacitor ? 10.0 ?F, and a battery of EMF ℇ 5.00 V as shown in the figure. Assume the capacitor is a parallel-plate capacitor, and it starts off initially uncharged. (a) (2 points) At time ? 0, the switch ? is closed. Immediately after the switch is closed, what is the current through resistor ? ? (b) (3 points) After a very long time, what is the potential at point A shown in the figure, relative to the negative terminal of the battery? (c) (3 points) After a very long time, what is the charge on the capacitor? (d) (4 points) After a very long time, a dielectric of constant ? 1.50 is inserted into the capacitor. After waiting another very long time, what is the potential across the capacitor, and how much charge is stored on the capacitor? (e) (4 points) A very long time after the dielectric was inserted, now switch ? is opened (so it looks like the figure). What is the current through ? immediately after the switch is opened? Assume the dielectric remains inside. (f) (4 points) How much time will pass from opening switch ? until the capacitor has half the charge it had before the switch was opened? (2a) A singly-charged positive ion has velocity ? 5.00x105 m/s �̂� moving right. The ion has charge 1.602x10-19 C. (i) (3 points) The ion enters a region of uniform electric and magnetic fields. The electric field is ? 1000 V/m ̂ pointing downward. If the ion is to pass through this region undeflected, then what is the magnitude and direction of the magnetic field? (ii) (3 points) Next the ion enters a second region with only a uniform magnetic field ? 0.750 T ? pointing out of the page toward you. What is the mass of the ion if it moves in semi-circle with radius R= 0.277 m? (iii) (2 points) How much time did it take the ion to move around the semi-circle? y z (out of page) x (2b) A loop of wire is bent into a semi-circle with radius R= 0.400 m and carries a current of 10.0 Amps. The loop is in a region of uniform magnetic field ? 0.100 T ̂pointing downward. At the instant when the plane of the loop is in the x-y plane as shown in the figure: (i) (4 points) What is the force on the straight section of wire of length 2R? (ii) (2 points) What is the force on the semi-circular section of the loop? (iii) (3 points) What is the magnetic moment vector of the loop? (iv) (3 points) What is the magnetic torque acting on the loop? R I ? I y z (out of page) x (3) A wire is bent into a triangular loop with one corner cut off and replaced by a segment of a circle of radius R and with point P as its center, as shown in the figure. The point P is at the point where the missing corner of the triangle would have been. The triangle is a right-angled triangle and both short sides of the triangle have length 2R. The loop is flat and lies in the x-y plane. A current I is flowing clockwise around the loop as seen from the positive z-axis, which comes out of the page. (a) (2 points) What is the magnitude and direction of the magnetic field due to the straight side AB at point P? (b) (2 points) What is the magnitude and direction of the magnetic field due to the straight side CD at point P? (c) (7 points) What is the magnitude and direction of the magnetic field due to circular segment side BC at point P? (d) (7 points) What is the magnitude and direction of the magnetic field due to the straight segment side DA at point P? (e) (2 points) What is the total magnetic field at point P? Constants ? 1 4??0 8.99 10 Nm /C ?0 8.85 10 C / Nm ?0 4? 10 N/A ? 1 ?0?0 3.00 10 m/s ? 1.60 10 C Proton/Electron Charge ? 9.11 10 kg Electron Mass Electric Force ? ? ? 4??0? ??? ? ?? ? ?? ? ? ? ? ? ∙ ? Electric Field ? 1 4??0 ? ? ? ? 1 4??0 ? ? ?? ? 1 4??0 ?? ? ? Φ ? ∙ ?? ? ? ?0 Electric Potential ? ? ? 4??0? ? ?? ? ? 4??0? ? 1 4??0 ? ? ? 1 4??0 ?? ? ? ? ? ∙ ?? ? ?? ?? , ?? ?? , ?? ?? Capacitance ? ? ? ? ?0? ? ? ??0 ? ? Parallel 1 ? 1 ? Series ? 1 2 ? ? 1 2 ?? 1 2 ?? ? 1 2 ?0? Circuits ? ?? ?? ?|?|? ? ? ??? ? ?? ? ?? ? ?? ? ? ?? ? 0 ? 0 Kirchhoff s Rules 1 ? 1 ? Parallel ? ? Series Magnetic Force ? ?? ? ? ?? ? ? ?? ? ? ? ? ? ∙ ? Magnetic Field ? ?0 4? ?? ? ? ?? ?0 4? ??? ? ? Scalar and Vector Products ? ∙ ? ? ? cos? ? ? ? ? ? ? ? ? ? ? sin? ? ? ? ? ? ? , ? ? ? ? , ? ? ? ? RC-Circuit ? ? ? 1 ? / Charging Capacitor ? ? ?0? / Discharging Capacitor ? ? ?0? / Both ? ?? Geometry ? 4?? ? 4 3 ?? ? 2??? ? ?? Mechanics ? ?? ? ? ? ? ? ∙ Δ? ? ? ? ? ? ? ?0 ?0? 1 2 ?? ? ?0 ?? Derivatives ? ?? ? ? ? ? ?? sin ?? ? cos ?? ? ?? cos?? ? sin ?? Integrals ?? √? ? ln ? ? ? ? ?? √? ? ? ? ?? ? ? 1 ? tan ? ? ? ?? ? ? 1 2 ln ? ? ?? ? ? 1 ? ? √? ? ? ?? ? ? 1 √? ? sin? ?? cos ? cos ? ?? sin?