The magnitude J of the current density in a certain lab wire with a circular cross section of radius R = 2.00 mm is given by , with J in amperes per square meter and radial distance r in meters. What is the current through the outer section bounded by r = 0.900 R and r = R ?
Current through the outer section bounded by radial distance r = 0.900 R and r = R is .
a) Current density,
b) Radius of the wire,
c) Bounded radial distances, and
The current is the rate of flow of charges per unit of time. The current density is the current per unit cross-section area of the rate of flow of charges per unit time per unit area.
We can use the relation between the current and the current density to find the current through the outer section bounded between the two radii.
The equation of the current flowing through a small area, ...(i)
Here, i is current, is current density, is the area of cross-section.
The cross-sectional area of the circle, ...(ii)
We have, the given value of the current density as:
The differential cross-sectional area value using equation (ii) can be given as follows:
Substituting these above values in the equation (i), we can get the contained current within the width of the concentric ring assuming is directed along the wire for the radial distance varying from to r = R as follows:
Hence, the value of the current is .
Figure-a shows a rod of resistive material. The resistance per unit length of the rod increases in the positive direction of the x axis. At any position x along the rod, the resistance dR of a narrow (differential) section of width dx is given by dR = 5.00 xdx , where is in ohms and x is in meters. Figure-b shows such a narrow section. You are to slice off a length of the rod between x = 0 and some position x = L and then connect that length to a battery with potential difference V = 5.0V (Figure-c). You want the current in the length to transfer energy to thermal energy at the rate of 200 W . At what position x = L should you cut the rod?
A certain cylindrical wire carries current. We draw a circle of radius r around its central axis in figure-a to determine the current i within the circle. Figure-b shows current i as a function of r2. The vertical scale is set by ,and the horizontal scale is set by, . (a) Is the current density uniform? (b) If so, what is its magnitude?
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