Two long straight parallel conductors 10 cm apart, carry currents of 5A each in the same direction. Then the magnetic induction at a point midway between them is

Options

A. `4xx10^(_5)T`
B. `2xx10^(-5)T`
C. `10^(-5)T`
D. zero

Answer

To solve the problem of finding the magnetic induction at a point midway between two long straight parallel conductors carrying currents in the same direction, we can follow these steps:

### Step 1: Understand the Setup
We have two long straight parallel conductors that are 10 cm apart, each carrying a current of 5 A in the same direction. We need to find the magnetic induction (magnetic field) at a point that is midway between these two conductors.

**Hint:** Visualize the arrangement of the conductors and the point of interest.

### Step 2: Determine the Distance from the Conductors
Since the conductors are 10 cm apart, the distance from each conductor to the midpoint is half of that distance:
\[ r = \frac{10 \, \text{cm}}{2} = 5 \, \text{cm} = 0.05 \, \text{m} \]

**Hint:** Convert all measurements to SI units for consistency.

### Step 3: Use the Formula for Magnetic Induction
The magnetic induction \( B \) due to a long straight conductor carrying current \( I \) at a distance \( r \) is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
where \( \mu_0 \) (the permeability of free space) is \( 4\pi \times 10^{-7} \, \text{T m/A} \).

**Hint:** Remember to substitute the values correctly into the formula.

### Step 4: Calculate the Magnetic Induction from Each Conductor
For each conductor, substituting \( I = 5 \, \text{A} \) and \( r = 0.05 \, \text{m} \):
\[ B_1 = \frac{\mu_0 \cdot 5}{2 \pi \cdot 0.05} \]
\[ B_1 = \frac{(4\pi \times 10^{-7}) \cdot 5}{2 \pi \cdot 0.05} \]
\[ B_1 = \frac{4 \times 10^{-7} \cdot 5}{0.1} \]
\[ B_1 = \frac{20 \times 10^{-7}}{0.1} = 2 \times 10^{-6} \, \text{T} \]

**Hint:** Calculate the magnetic field contribution from one conductor first.

### Step 5: Determine the Direction of the Magnetic Fields
The magnetic field \( B_1 \) due to the first conductor will be directed into the page (or screen), while the magnetic field \( B_2 \) due to the second conductor will be directed out of the page (or screen).

**Hint:** Use the right-hand rule to determine the direction of the magnetic fields.

### Step 6: Calculate the Total Magnetic Induction
Since the magnitudes of \( B_1 \) and \( B_2 \) are equal but their directions are opposite, the total magnetic induction \( B \) at the midpoint will be:
\[ B = B_1 - B_2 = 2 \times 10^{-6} - 2 \times 10^{-6} = 0 \]

**Hint:** Remember to consider the vector nature of magnetic fields when adding them.

### Conclusion
The magnetic induction at the point midway between the two conductors is:
\[ B = 0 \, \text{T} \]

### Final Answer
The correct option is 0.

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