Electric charge flows through a wire, which naturally has a cylindrical cross section. Suppose the charge has a current density given by J = A/r, where A is a constant and r is the distance to the wire's central axis.
How much current flows through the wire if its radius is R?
To find the current flowing through the wire, we need to calculate the total charge passing through a cross-section of the wire per unit of time.
First, we need to determine the relationship between current density J and current I. Current density J is defined as the amount of current passing through a given area per unit of area. Mathematically, we have:
J = I/A,
where I is the current and A is the cross-sectional area of the wire.
Given that the current density J is given by J = A/r, we can replace J in the equation above:
A/r = I/A.
To find the current I, we need to consider the cross-sectional area A of the wire. The cross-sectional area of a cylinder is given by A = πr^2, where r is the radius.
Substituting this value of A into the equation A/r = I/A, we get:
(πr^2)/r = I/(πr^2).
Simplifying the equation, we have:
πr = I/(πr^2).
Now, let's solve for the current I. Multiply both sides of the equation by (πr^2) to isolate I on one side:
I = πr^3.
Therefore, the current flowing through the wire with radius R is given by I = πR^3.
So, if you know the radius R of the wire, you can calculate the current flowing through it using the formula I = πR^3.