A long wire carrying a 6.0 A current perpendicular to the xy-plane intersects the x-axis at x=−2.0cm. A second, parallel wire carrying a 2.0 A current intersects the x-axis at x=+2.0cm. At what point on the x-axis is the magnetic field zero if the two currents are in the same direction?
At what point on the x-axis is the magnetic field zero if the two currents are in opposite directions?
Well, it seems like those wires are having quite the magnetic field party! Let's figure out where things get interesting.
If the two currents are in the same direction, the magnetic fields created by the wires will add up. Now, we know that the magnetic field caused by a straight wire is inversely proportional to the distance from the wire. So, at some point on the x-axis between -2.0 cm and +2.0 cm, the magnetic field will reach zero when the magnitudes of the fields created by the two wires cancel each other out.
Now, to find that point, we need to use a bit of math (I promise, no clown math involved!). Since the wires are parallel, the net magnetic field will be zero at a point equidistant from both wires. Since the wires intersect the x-axis at -2.0 cm and +2.0 cm respectively, the midpoint between them would be at 0 cm. So, the magnetic field will be zero at x=0 cm if the two currents are in the same direction.
Now, let's flip the script and imagine the two currents are in opposite directions. This means the magnetic fields created by the wires will oppose each other. Similar to before, at some point on the x-axis, the magnitudes of the fields created by the two wires will cancel each other out.
By using the same midpoint logic, we find that the magnetic field will be zero at x=0 cm again if the two currents are in opposite directions.
So, in both cases, the magnetic field will be zero at x=0 cm. The wires really do like to zero out the fun!
To find the point on the x-axis where the magnetic field is zero, we can apply the Biot-Savart Law, which states that the magnetic field produced by a current-carrying wire is directly proportional to the current and inversely proportional to the distance from the wire.
1. When the currents are in the same direction:
Let's assume the distance from the wire carrying 6.0 A current to the point on the x-axis where the magnetic field is zero is "x". According to the Biot-Savart Law, the magnetic field produced by the wire is given by:
B1 = (μ0 * I1) / (2π * x)
where μ0 is the permeability of free space, and I1 is the current in the wire.
Similarly, for the wire carrying 2.0 A current, the magnetic field at the same point on the x-axis is given by:
B2 = (μ0 * I2) / (2π * (0.02 - x))
To find the point where the magnetic field is zero, we set B1 = B2 and solve for x:
(μ0 * I1) / (2π * x) = (μ0 * I2) / (2π * (0.02 - x))
Simplifying the equation:
6.0 / x = 2.0 / (0.02 - x)
Cross-multiplying:
2.0x = 6.0(0.02 - x)
Expanding and solving for x:
2.0x = 0.12 - 6.0x
8.0x = 0.12
x = 0.015 m = 1.5 cm
Therefore, the magnetic field is zero at a point 1.5 cm away from the wire carrying the 6.0 A current.
2. When the currents are in opposite directions:
Using the same approach as above, we set up the equation:
B1 = (μ0 * I1) / (2π * x) = -((μ0 * I2) / (2π * (0.02 - x)))
Simplifying the equation:
6.0 / x = -2.0 / (0.02 - x)
Cross-multiplying:
-2.0x = 6.0(0.02 - x)
Expanding and solving for x:
-2.0x = 0.12 - 6.0x
-4.0x = 0.12
x = -0.03 m = -3.0 cm
Therefore, the magnetic field is zero at a point 3.0 cm away from the wire carrying the 6.0 A current when the currents are in opposite directions.
To find the point on the x-axis where the magnetic field is zero, we can apply Ampere's law. Ampere's law states that the magnetic field around a closed loop is proportional to the current passing through that loop.
1. When the currents are in the same direction:
Let's consider a circular loop centered on the x-axis and having a radius of r. The magnetic field B along the circular loop at any point is given by the equation:
B = (μ₀ * I₁) / (2 * Ï€ * r) + (μ₀ * I₂) / (2 * Ï€ * (d - r))
Where:
- μ₀ is the permeability of free space (constant)
- I₁ is the current passing through the first wire (6.0 A)
- I₂ is the current passing through the second wire (2.0 A)
- d is the separation between the two wires (4.0 cm = 0.04 m)
To find the point where the magnetic field becomes zero, we set B = 0 and solve the equation:
0 = (μ₀ * 6.0) / (2 * Ï€ * r) + (μ₀ * 2.0) / (2 * Ï€ * (0.04 - r))
Simplifying the equation gives:
3 / r + 1 / (0.04 - r) = 0
Solving this equation will give you the value of r, which represents the distance from the origin to the point where the magnetic field is zero.
2. When the currents are in opposite directions:
Similarly, we consider a circular loop as before and derive the equation for the magnetic field B along the loop:
B = (μ₀ * I₁) / (2 * Ï€ * r) - (μ₀ * I₂) / (2 * Ï€ * (d - r))
Setting B = 0 and solving the equation will give you the distance r from the origin where the magnetic field is zero.
By solving these equations, you can find the points on the x-axis where the magnetic field is zero for both cases.