Two very long, straight, parallel wires separated by distance d carry currents of magnitude I in opposite directions. Find the magnetic field at a point equidistant from the lines, with a distance 2d from each line.
just no
To find the magnetic field at a point equidistant from the two wires, we can use 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.
The magnetic field produced by a single wire can be calculated using the equation:
\[ B = \frac{{\mu_0 \cdot I}}{2\pi \cdot r} \]
Where:
- B is the magnetic field strength,
- \(\mu_0\) (mu naught) is the permeability of free space (a constant equal to \(4\pi \times 10^{-7}\) Tm/A),
- I is the current in the wire, and
- r is the distance from the wire.
In this case, we have two parallel wires carrying currents of magnitude I in opposite directions. The wires are separated by distance d, and we want to find the magnetic field at a point that is equidistant from the wires, with a distance of 2d from each wire.
To find the magnetic field at the desired point, we need to calculate the magnetic fields produced by each wire separately and then add them together, taking into account the direction of the currents.
Let's calculate the magnetic field produced by the wire to the left of the point:
\[ B_{\text{left}} = \frac{{\mu_0 \cdot I}}{2\pi \cdot r_{\text{left}}} \]
Where \( r_{\text{left}} \) is the distance from the wire to the desired point.
Since the point is equidistant from both wires, \( r_{\text{left}} = 2d \).
Similarly, let's calculate the magnetic field produced by the wire to the right of the point:
\[ B_{\text{right}} = \frac{{\mu_0 \cdot I}}{2\pi \cdot r_{\text{right}}} \]
Where \( r_{\text{right}} \) is the distance from the wire to the desired point.
Again, since the point is equidistant from both wires, \( r_{\text{right}} = 2d \).
Since the currents in the two wires are in opposite directions, the magnetic fields they produce will also have opposite directions. Therefore, we subtract the magnetic field produced by the wire to the right from the magnetic field produced by the wire to the left:
\[ B = B_{\text{left}} - B_{\text{right}} \]
Substituting the above equations for \( B_{\text{left}} \) and \( B_{\text{right}} \), we get:
\[ B = \left( \frac{{\mu_0 \cdot I}}{2\pi \cdot r_{\text{left}}} \right) - \left( \frac{{\mu_0 \cdot I}}{2\pi \cdot r_{\text{right}}} \right) \]
Simplifying further:
\[ B = \frac{{\mu_0 \cdot I}}{2\pi} \left( \frac{1}{r_{\text{left}}} - \frac{1}{r_{\text{right}}} \right) \]
Substituting the values of \( r_{\text{left}} \) and \( r_{\text{right}} \):
\[ B = \frac{{\mu_0 \cdot I}}{2\pi} \left( \frac{1}{2d} - \frac{1}{2d} \right) \]
Since the two terms in the brackets cancel out, we are left with:
\[ B = 0 \]
Therefore, the magnetic field at a point equidistant from the two wires and with a distance of 2d from each wire is zero.