The wire in the drawing carries a current of 14 A. Suppose that a second long, straight wire is placed right next to this wire. The current in the second wire is 33 A. Use Ampere's law to find the magnitude of the magnetic field at a distance of r = 1.0 m from the wires when the currents are (a) in the same direction and (b) in opposite directions.
Well, well, well, looks like we've got some electrifying wires here! Let me put on my magnetic clown shoes and help you out with Ampere's law.
(a) When the currents are in the same direction, the magnetic field at a distance of 1.0 m can be calculated using Ampere's law:
B = (μ₀ * I₁) / (2π * r₁) + (μ₀ * I₂) / (2π * r₂)
Substituting values, we have:
B = (4π * 10^-7 T·m/A * 14 A) / (2π * 1.0 m) + (4π * 10^-7 T·m/A * 33 A) / (2π * 1.0 m)
Simplifying the equation, we find:
B = (56 * 10^-7 T) / 1.0 m + (132 * 10^-7 T) / 1.0 m
B = 56 * 10^-7 T/m + 132 * 10^-7 T/m
B = 188 * 10^-7 T/m
That's quite a magnetic field, my friend!
(b) Now, let's explore what happens when the currents are in opposite directions. Brace yourself for some magnetic opposites!
The formula remains the same, but the second current becomes negative because they're going in opposite directions.
B = (μ₀ * I₁) / (2π * r₁) + (μ₀ * (-I₂)) / (2π * r₂)
Substituting values and taking the negative sign into account:
B = (4π * 10^-7 T·m/A * 14 A) / (2π * 1.0 m) + (4π * 10^-7 T·m/A * (-33 A)) / (2π * 1.0 m)
Simplifying the equation, we find:
B = (56 * 10^-7 T) / 1.0 m - (132 * 10^-7 T) / 1.0 m
B = 56 * 10^-7 T/m - 132 * 10^-7 T/m
B = -76 * 10^-7 T/m
Oh dear, the magnetic field gets a bit gloomy when the currents are in opposite directions. Remember, magnets can be fickle!
So, there you have it! For the same direction, it's 188 * 10^-7 T/m, and for opposite directions, it's -76 * 10^-7 T/m. Hope this helps lighten up your day!
To find the magnitude of the magnetic field using Ampere's law, we need to calculate the magnetic field produced by each wire separately and then combine them.
Ampere's law states that the line integral of the magnetic field around a closed loop is equal to μ₀ times the total current passing through the loop.
Mathematically, Ampere's law can be expressed as:
∮ B · dl = μ₀ * I_enc
Where:
- B is the magnetic field vector
- dl is an infinitesimal vector element along the path of integration
- μ₀ is the permeability of free space (4π × 10^-7 T·m/A)
- I_enc is the total current enclosed by the closed loop of integration
Let's calculate the magnetic field produced by each wire separately:
For the wire with a current of 14 A:
The total current enclosed by the loop is 14 A since the loop encloses only this wire.
Let's call this magnetic field B₁.
For the wire with a current of 33 A:
Again, the total current enclosed by the loop is 33 A since the loop encloses only this wire.
Let's call this magnetic field B₂.
Now, let's consider the two cases mentioned:
(a) When the currents are in the same direction:
In this case, we need to add the magnetic fields produced by each wire together.
The total magnetic field at a distance of 1.0 m from the wires is given by:
B_total = B₁ + B₂
(b) When the currents are in opposite directions:
In this case, the magnetic fields produced by each wire are in opposite directions and must be subtracted.
The total magnetic field at a distance of 1.0 m from the wires is given by:
B_total = |B₁ - B₂|
Use the formula for the magnetic field produced by a long straight wire at a distance r:
B = (μ₀ * I) / (2π * r)
Let's plug in the values and calculate:
For the wire with a current of 14 A:
B₁ = (μ₀ * I₁) / (2π * r)
For the wire with a current of 33 A:
B₂ = (μ₀ * I₂) / (2π * r)
Now, we can substitute these values into the formulas above and calculate B_total in each case.