# UY1: Magnetic Field & Force Between Parallel Conductors

### 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 $\mu_{0} = 4 \pi \times 10^{-7} \text{T m A}^{-1}$.

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