a) Two conductors X and Y carrying currents I 1 and I 2 are placed in air at a distance r apart as shown in figure. [figure in the original paper] i) Find the magnitude of force between them. [2]…

Answer

i) Magnitude of force between the conductors

The force per unit length between two parallel current-carrying conductors placed in air (or vacuum) is given by Ampère’s law:

$$\frac{F}{L}=\frac{{\mu }_0}{4\pi }\frac{I_1{I}_2}{r}$$

Where:

• $F$ = Force between the conductors (in Newtons, N)
• $L$ = Length of the conductors (in meters, m)
• ${\mu }_0$ = Permeability of free space = $4\pi \times 10^{-7}$ T·m/A
• $I_1,I_2$ = Currents in the conductors (in Amperes, A)
• $r$ = Distance between the conductors (in meters, m)

Step-by-step calculation:

1. Substitute ${\mu }_0=4\pi \times 10^{-7}$ T·m/A into the formula: $$\frac{F}{L}=\frac{4\pi \times 10^{-7}}{4\pi }\frac{I_1{I}_2}{r}$$
2. Simplify: $$\frac{F}{L}=10^{-7}\frac{I_1{I}_2}{r}$$
3. If the question expects a general expression, the force per unit length is: $$\boxed{\frac{F}{L}=10^{-7}\frac{I_1{I}_2}{r}\text{ N/m}}$$ If numerical values for $I_1,I_2,$ and $r$ were given, substitute them here (e.g., if $I_1=I_2=1$ A and $r=5$ m, then $F/L=4\times 10^{-5}$ N/m, as shown in the figure).

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ii) Nature of the force

The force between two parallel current-carrying conductors is:

• Attractive if the currents flow in the same direction (as shown in the figure).
• Repulsive if the currents flow in opposite directions.

Since the figure (assumed to show currents in the same direction) is not provided, we assume the standard case where currents are parallel and in the same direction. Thus, the force is attractive.

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iii) Definition of one ampere based on force

One ampere is defined as:

"The constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to $2\times 10^{-7}$ N per meter of their length."

In other words: $$\boxed{1\text{ A}=\text{Current in each conductor such that }\frac{F}{L}=2\times 10^{-7}\text{ N/m when }r=1\text{ m}}$$