Question

Two parallel long straight conductors are placed at right angles to a meter scale at the two centimeter and six centimeter marks they carry currents I and 3 I respectively in the same direction the resultant magnetic field due to them is 0

Answer 1

3 cm mark

Answer 2

To find the point where the resultant magnetic field is zero, we need to consider the magnetic field produced by each conductor and their directions.

1. Magnetic Field due to a Long Straight Conductor: The magnitude of the magnetic field ($B$) at a distance ($r$) from a long straight conductor carrying current ($I$) is given by: $B = \frac{\mu_0 I}{2 \pi r}$ The direction of the magnetic field is given by the right-hand thumb rule.

2. Setup and Directions: Let the first conductor (carrying current $I$) be at $x_1 = 2$ cm and the second conductor (carrying current $3I$) be at $x_2 = 6$ cm on the meter scale. Both currents are in the same direction (e.g., out of the page).

   • Region to the left of 2 cm (x < 2 cm): The magnetic field due to both conductors will be in the same direction (e.g., downwards, if current is out of page). Thus, the fields add up, and the resultant field cannot be zero.
   • Region between 2 cm and 6 cm (2 cm < x < 6 cm): The magnetic field due to the first conductor (at 2 cm) will be in one direction (e.g., upwards). The magnetic field due to the second conductor (at 6 cm) will be in the opposite direction (e.g., downwards). Since the fields are in opposite directions, there is a possibility for them to cancel out and result in a zero magnetic field.
   • Region to the right of 6 cm (x > 6 cm): The magnetic field due to both conductors will be in the same direction (e.g., upwards). Thus, the fields add up, and the resultant field cannot be zero.

   Therefore, the point where the resultant magnetic field is zero must lie between the two conductors.

3. Calculating the Position: Let the point where the resultant magnetic field is zero be at position $x$ cm on the meter scale.

   • Distance from the first conductor ($I$) is $r_1 = (x - 2)$ cm.
   • Distance from the second conductor ($3I$) is $r_2 = (6 - x)$ cm.

   For the resultant magnetic field to be zero, the magnitudes of the magnetic fields produced by each conductor at point $x$ must be equal: $B_1 = B_2$ $\frac{\mu_0 I_1}{2 \pi r_1} = \frac{\mu_0 I_2}{2 \pi r_2}$ Substituting the given values ($I_1 = I$, $I_2 = 3I$): $\frac{\mu_0 I}{2 \pi (x - 2)} = \frac{\mu_0 (3I)}{2 \pi (6 - x)}$ Cancel out common terms ($\frac{\mu_0 I}{2 \pi}$): $\frac{1}{x - 2} = \frac{3}{6 - x}$ Now, solve for $x$: $6 - x = 3(x - 2)$ $6 - x = 3x - 6$ $6 + 6 = 3x + x$ $12 = 4x$ $x = \frac{12}{4}$ $x = 3 \text{ cm}$

   This position $x = 3$ cm lies between 2 cm and 6 cm, which is consistent with our analysis of directions.

The resultant magnetic field due to them is 0 at the 3 cm mark.