The wires leading to and from a 0.12--diameter lightbulb filament are 1.5 in diameter. The wire to the filament carries a current with a current density of 4.0×105 . what is the current density in the filament?
To find the current density in the filament, we need to determine the current passing through the wire leading to the filament and the area of the filament.
The current density (J) is defined as the current (I) passing through a conductor divided by the cross-sectional area (A) of the conductor.
Given:
- Diameter of the filament wire (d_f) = 0.12 in
- Diameter of the wire leading to the filament (d_l) = 1.5 in
- Current density in the wire leading to the filament (J_l) = 4.0 × 10^5 A/m^2
First, let's convert the given measurements into meters.
1 inch = 0.0254 meters
d_f = 0.12 in * 0.0254 m/in = 0.003048 m
d_l = 1.5 in * 0.0254 m/in = 0.0381 m
Next, we need to find the current passing through the wire leading to the filament (I_l).
The current density (J_l) is given as J_l = 4.0 × 10^5 A/m^2.
We can use the equation J = I / A to solve for I:
I_l = J_l * A_l
The cross-sectional area of the wire leading to the filament (A_l) can be obtained using the formula for the area of a circle:
A_l = π * (d_l/2)^2
Substituting the values, we have:
A_l = π * (0.0381/2)^2
A_l = 0.001133 m^2 (rounded to 4 decimal places)
Now, we can substitute the values into the equation I_l = J_l * A_l:
I_l = 4.0 × 10^5 A/m^2 * 0.001133 m^2
I_l = 453.2 A (rounded to the nearest whole number)
Therefore, the current density in the filament is 453.2 A.
To find the current density in the filament, we need to determine the current passing through the wire to the filament and the cross-sectional area of the filament.
Step 1: Calculate the current passing through the wire to the filament.
We are given the current density in the wire leading to the filament, which is 4.0×10^5 A/m^2. However, we do not know the diameter of this wire.
Step 2: Calculate the cross-sectional area of the wire to the filament.
The cross-sectional area of a wire is given by the formula: A = πr^2, where A is the cross-sectional area and r is the radius of the wire.
Given:
Diameter of the wire = 1.5 in
Radius = diameter/2 = 1.5/2 = 0.75 in
Step 3: Convert the units to match the current density.
The current density is given in A/m^2, so we need to convert the units of the wire's cross-sectional area from inches to meters.
1 inch = 0.0254 meters (conversion factor)
Radius in meters = 0.75 in * 0.0254 m/in = 0.01905 m
Step 4: Calculate the cross-sectional area of the wire to the filament.
Using the formula A = πr^2:
A = π * (0.01905 m)^2 ≈ 0.001140 m^2
Step 5: Find the current passing through the wire to the filament.
We can find the current passing through the wire by multiplying the current density by the cross-sectional area:
Current passing through the wire = Current density * Cross-sectional area
Current passing through the wire = 4.0×10^5 A/m^2 * 0.001140 m^2 ≈ 456 A
Step 6: Calculate the current density in the filament.
The current passing through the wire to the filament is the same as the current passing through the filament itself. Therefore, the current density in the filament is also 4.0×10^5 A/m^2.
Answer: The current density in the filament is 4.0×10^5 A/m^2.