UY1: Magnetic Field & Force Between Parallel Conductors

Magnetic Field Of Two Wires

Magnetic field of two wires

Figure shows an end view of two long, straight, parallel wires perpendicular to the xy-plane, each carrying a current I but in opposite directions. Find the magnitude and direction of $\vec{B}$ at point $P_{1}$. Find the magnitude and direction of $\vec{B}$ at any point on the x-axis to the right of wire 2 in terms of the x-coordinate of the point.

The magnetic field of a long straight current carrying conductor is given by:

$$B = \frac{\mu_{0}I}{2 \pi r}$$

Hence, the magnetic field at point $P_{1}$ is given by:

$$\begin{aligned} \vec{B}_{P_{1}} &= \frac{\mu_{0}I}{2 \pi} \left( -\frac{1}{2d} + \frac{1}{4d} \right) \hat{j} \\ &= -\frac{\mu_{0}I}{8 \pi d} \hat{j} \end{aligned}$$

The magnetic field for the second part is given by:

$$\begin{aligned} \vec{B}_{x} &= \frac{\mu_{0}I}{2 \pi} \left( \frac{1}{x+d} – \frac{1}{x-d} \right) \hat{j} \\ &= – \frac{\mu_{0}Id}{\pi \left(x^{2}-d^{2} \right)} \hat{j} \end{aligned}$$

Force Between Parallel Conductors

Consider segments of two long, straight parallel conductors separated by a distance r and carrying currents I and I’, respectively, in the same direction.

Each of the conductor produces a magnetic field that is given by: (where I can be I or I’ – depending on the current carried by the conductor)

$$B = \frac{\mu_{0}I}{2 \pi r}$$

The force experienced by one of the conductor due to the magnetic field produced by the other conductor is given by:

$$\begin{aligned} \vec{F} &= I’ \vec{L} \times \vec{B} \\ &= I’ L B \\ &= \frac{\mu_{0}II’L}{2\pi r} \\ \frac{F}{L} &= \frac{\mu_{0}II’}{2 \pi r} \end{aligned}$$

Two parallel conductors carrying current in the same direction attract each other. Parallel conductors carrying currents in opposite directions repel each other.

Note: It will be easy to see this if you do it via the vectors method instead of only computing the magnitude of the force as is done above.

SI Definition Of The Ampere:

One ampere is that unvarying current that, if present in each of two parallel conductors of infinite length and one meter apart in empty space, causes each conductor to experience a force of exactly $2 \times 10^{-7}$ newtons per meter of length.

If you put $I = 1 \text{A}$ and $L = 1 \text{ m}$ into the equation for force between two parallel conductor, you will arrive at $\mu_{0} = 4 \pi \times 10^{-7} \text{T m A}^{-1}$.


Next: Magnetic Field Of A Circular Current Loop

Previous: Magnetic Field Of A Straight Current Carrying Conductor

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