home / study / science / advanced physics / questions and answers / in the figur

Question

home / study / science / advanced physics / questions and answers / in the figure, a compass sits 1.0 cm above a wire ...

Question: In the figure, a compass sits 1.0 cm above a wire ...

Bookmark

In the figure, a compass sits 1.0 cm above a wire in a circuit containing a 1.0 F capacitor charged to 50 V , a 2.0 resistor, and an open switch. Initially, the compass is lined up with the earth’s magnetic field. The switch is then closed, so there is a current in the circuit. The switch remains closed until the capacitor has completely discharged.

a. At the position of the compass, what is the direction of the magnetic field due to the current in the wire after the switch is closed? (Assume that the compass is only affected by the field created by the current in the wire that is directly below it.)

b. At the position of the compass, what is the magnitude of the magnetic field due to the current in the wire at the following times after the switch is closed? Fill in the table. How does this compare with the magnitude of the field of the earth? Use 50 µT for the magnitude of the earth’s field.

c. Describe how the compass orientation changes right after the switch is closed, and how the compass orientation changes for the next ten seconds.

Explanation / Answer

a) The magnetic field (H) due to a current (I) in a long straight wire is given by H = I/(2*pi*r) where r is the distance of the point of interest from the axis of the wire. The units of H are amps/meter. In your question, at the moment the switch is closed I will be Io = V/R = 4.9A, assuming that the total resistance in the circuit (including the long wire) is 1 ohm. This is a somewhat simplistic calculation, since, ideally the wire should be infinitely long, in which case the resistance would be infinite, and the current zero. However, using 1 Ohm we get H = 4.9/(2*pi*1.1* 10^-2) = 70.89A/m

b) the quantity measured in Teslas (T) is magnetic flux density, not magnetic field. To convert the given value of flux density to a corresponding value of field strength you divide by the permeability of air (or vacuum) to get H = B/uo = 5*10^-5/1.26*10^-6 = 39.8A/m. The field due to the current is thus 1.78 times as strong as the earth's field.

c) It is not possible to answer this part of the question for several reasons -

1) If the field due to the wire were constant, the compass needle would align itself with the field which is the vector sum of the earth's field, and that due to the current in the wire (Hw). The direction of the latter is normal to the direction of the current, but we are not told the direction of the current (i.e. that of the wire) relative to the direction of the earth's field, so we cannot perform the required vector addition.

2) The current in the wire decreases rapidly with time as the capacitor discharges. The time-scale for these changes is of the order of the time constant of the RC product - this is 1s. The inertia of the compass needle must be very small if it is to follow these changes accurately, and it must be critiaclly damped, but we are not told anything about the inertial properties or the damping of the needle. Thus even if we were able to determine the direction of Hw, we still would not be able to say how the needle would respond to the rapidly changing value and direction of Hw. The variation of I with time is given by this equation, from which the magnitude of H as a function of time can be determined: I(t) = Io*(1-e^(-t/RC)) where e is the base of the natural logs, R is the total resistance, and C is the capacitance.

It is likely that the compass needle will indicate a brief jerk to one side or the other, followed by a return to its original position, and it might well oscillate about its original position, executing a damped SHM motion. But it would not respond at all if Hw is in the same direction as the earth's field. If Hw were opposed to the earth's field, the needle might make a complete rotation before executing a series of (initially) large-amplitude damped oscillations about its original position.

Related Questions

Navigate

• Browse (All)
• Browse H
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.