High-frequency signals are often transmitted along a coaxial cable, such as the one shown in FIGURE P26.66. 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 (i.e., at ). An insulating material fills the space between them, and an insulating plastic coating goes around the outside.
a. Find an expression for the capacitance per meter of a coaxial cable. Assume that the insulating material between the cylinders is air.
b. Evaluate the capacitance per meter of a cable having and
a. The expression for the capacitance per meter of a coaxial cable is
b. The capacitance of a cable is
The signal is carried on a wire of radius
a. Applying Gauss's law for the cylinder with linear charge density , to get electric field outside of a cylinder as
Potential difference for two cylinders will be
Find the capacitance per unit length as
Hence, the expression for the capacitance per meter of a coaxial cable is
The signal is carried on a wire of radius and
In the given numerical case, we will have
Substitute the values,
Multiply the expression,
Hence, the capacitance of a cable is
A nerve cell in its resting state has a membrane potential of , meaning that the potential inside the cell is less than the potential outside due to a layer of negative charge on the inner surface of the cell wall and a layer of positive charge on the outer surface. This effectively makes the cell wall a charged capacitor. When the nerve cell fires, sodium ions,, flood through the cell wall to briefly switch the membrane potential to . Model the central body of a nerve cell-the soma-as a diameter sphere with a -thick cell wall whose dielectric constant is 9.0. Because a cell's diameter is much larger than the wall thickness, it is reasonable to ignore the curvature of the cell and think of it as a parallel-plate capacitor. How many sodium ions enter the cell as it fires?
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