A metal bar has a rectangular cross section 5.0cm by 10cm. The bar has a non-uniform conductivity, and as a result the current density increases linearly from 0 at the bottom to 0.10A/cm^2 at the top. Find the total current in the bar.
Hint: first determine the current dI through an infinitesimal area section dA.
To find the total current in the bar, we need to integrate the current density over the entire cross-sectional area of the bar. Let's first determine the current dI through an infinitesimal area section dA.
The current density (J) is defined as the current (I) per unit area (A). Since the current density increases linearly from 0 at the bottom to 0.10A/cm^2 at the top, we can express the current density as:
J = m*x + c,
where m is the slope of the linear increase (0.10A/cm^2 divided by the height of the bar), x is the position along the height of the bar, and c is the y-intercept or initial current density at the bottom (which is zero).
Now, let's consider an infinitesimal area element dA located at a height x from the bottom of the bar. The current dI passing through this infinitesimal area element is given by:
dI = J*dA.
Substituting the expression for J, we have:
dI = (m*x + c) * dA.
Next, let's express dA in terms of the dimensions of the rectangular cross section. The infinitesimal area element dA is given by the product of the dimensions in the x and y directions:
dA = dx * dy.
Since the cross section is a rectangle with dimensions of 5.0cm by 10cm, we can write:
dA = 5.0cm * dx.
Substituting this value for dA, we have:
dI = (m*x + c) * 5.0cm * dx.
Now, we can integrate this equation over the entire height of the bar (from 0 to the total height h) to obtain the total current (I):
I = ∫(0 to h) dI = ∫(0 to h) (m*x + c) * 5.0cm * dx.
Integrating with respect to x, we get:
I = ∫(0 to h) (m*x + c) * 5.0cm * dx
= [m*(x^2)/2 + c*x] * 5.0cm | (0 to h).
Evaluating this expression at the upper limit h and subtracting the evaluation at the lower limit 0, we obtain:
I = [m*(h^2)/2 + c*h] * 5.0cm.
Now, we can substitute the given values to calculate the total current.
Note: The value of h (total height) is not given in the problem statement, so you will need to use any additional information provided or make an assumption in order to calculate the current.