86. III High-frequency signals are often transmitted along a coaxial CALC cable, such as the one shown in FIGURE CP30.86. For example, the cable TV hookup coming into your home is a coaxial cable. The signal is carried on a wire of radius while the outer conductor of radius is grounded. A soft, flexible insulating material fills the space between them, and an insulating plastic coating goes around the outside.
a. Find an expression for the inductance per meter of a coaxial cable. To do so, consider the flux through a rectangle of length that spans the gap between the inner and outer conductors.
b. Evaluate the inductance per meter of a cable having and .
Part (a) The self-inductance per unit length of the cable is henries/m
Part (b) The inductance per meter is
Radio frequency AC are transmitted by means of cables over a large distance. A cable consists of two coaxial cylindrical conductors of radii and . The currents in the conductors are equal in magnitude but opposite in directions. Thus, there is no magnetic field outside these conductors. Thus, the cable is non-radiating. There will, however, be a magnetic flux between the two cylinders.
Let there be a current of I A through the conductors
The magnetic field B is at distance r from the axis and within the region between the two cylinders.
Here, the flux through the area between r and r+dr, and of unit length, is given by
Thus, total flux between the unit length of the cable is given by
Thus, self-inductance per unit length of the cable is henries/m
Here, , and
Thus, from the result of (a) above, the inductance per meter is
The inductance per meter is
I A 20-cm-long, zero-resistance slide wire moves outward, on zero-resistance rails, at a steady speed of in a magnetic field. (See Figure 30.26.) On the opposite side, a carbon resistor completes the circuit by connecting the two rails. The mass of the resistor is .
a. What is the induced current in the circuit?
b. How much force is needed to pull the wire at this speed?
c. If the wire is pulled for , what is the temperature increase of the carbon? The specific heat of carbon is.
11. a. Can you tell which of the inductors in FIGURE Q30.11 has the larger current through it? If so, which one? Explain.
b. Can you tell through which inductor the current is changing more rapidly? If so, which one? Explain.
c. If the current enters the inductor from the bottom, can you tell if the current is increasing, decreasing, or staying the same? If so, which? Explain.
A 10-turn coil of wire having a diameter of 1.0 cm and a resistance of is in a magnetic field, with the coil oriented for maximum flux. The coil is connected to an uncharged capacitor rather than to a current meter. The coil is quickly pulled out of the magnetic field. Afterward, what is the voltage across the capacitor?
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