The current density in a cylindrical conductor of radius R = 59.5 cm, varies as J(r) = J0r/R (in the region from zero to R), where J0 = 0.423 C/m2.
a) Find the magnitude of the magnetic field at r = 29.75 cm.
b) Find the magnitude of the magnetic field at r = 119 cm
To find the magnitude of the magnetic field at a given radius, we can use Ampere's Law. Ampere's Law relates the magnetic field around a closed loop to the current passing through the loop and the distance from the current.
Ampere's Law states that the line integral of the magnetic field along a closed path is equal to the product of the current enclosed by the path and the permeability of free space (μ₀).
Mathematically, Ampere's Law is expressed as:
∮ B • dl = μ₀ • I_enclosed
Let's solve each part of the question separately:
a) To find the magnitude of the magnetic field at r = 29.75 cm, we need to calculate the current enclosed at that radius.
The current enclosed (I_enclosed) at r = 29.75 cm is the total current that passes through the loop of radius 29.75 cm. We can determine this by integrating the current density function (J(r)) from 0 to 29.75 cm.
The formula for the total current enclosed is:
I_enclosed = ∫(J(r) • 2πr) dr
Plugging in the given current density function J(r) = J0r/R and integrating from 0 to 29.75 cm, we get:
I_enclosed = ∫(J0r/R • 2πr) dr
I_enclosed = ∫(J0 • 2πr²/R) dr
I_enclosed = J0 • (2π/R) • ∫(r²) dr, integrated from 0 to 29.75 cm
Evaluating the integral gives:
I_enclosed = J0 • (2π/R) • [r³/3] evaluated from 0 to 29.75 cm
I_enclosed = J0 • (2π/R) • [(29.75 cm)³/3]
Note that we convert the radius to meters (29.75 cm = 0.2975 m) to obtain the result in SI units.
Once we have the enclosed current, we can apply Ampere's Law to find the magnetic field magnitude.
∮ B • dl = μ₀ • I_enclosed
The line integral of the magnetic field (B) around a circular loop is simply the magnetic field (B) multiplied by the circumference of the loop (2πr) since the magnetic field is parallel to the loop everywhere.
So the equation becomes:
B • (2πr) = μ₀ • I_enclosed
Solving for B, we find:
B = (μ₀ • I_enclosed) / (2πr)
Plugging in the values, we have:
B = (μ₀ • J0 • (2π/R) • [(29.75 cm)³/3]) / (2π • 0.2975 m)
Now we can calculate the magnitude of the magnetic field at r = 29.75 cm.
b) To find the magnitude of the magnetic field at r = 119 cm, we can follow the same steps as in part a), but substitute 119 cm for r in the formulas.
I_enclosed = J0 • (2π/R) • [(119 cm)³/3]
B = (μ₀ • J0 • (2π/R) • [(119 cm)³/3]) / (2π • 1.19 m)
Now we can calculate the magnitude of the magnetic field at r = 119 cm.