Magnetic field due to a current carrying conductor at a point is,` B = (KI)/(r)` , where I is the current through the conductor and r is the distance of a point from the conductor. At 20 cm from the conductor for the 10 A current, magnetic field is 50 tesla. Keeping the distance same, what is the value of magnetic field in tesla) for 40 A current ? (K is the constant of proportionality).

Explanation

To solve the problem step by step, we will use the formula for the magnetic field due to a current-carrying conductor:

\[ B = \frac{KI}{r} \]

where:
- \( B \) is the magnetic field,
- \( K \) is the constant of proportionality,
- \( I \) is the current through the conductor,
- \( r \) is the distance from the conductor.

### Step 1: Identify the given values
From the problem, we have:
- Current in the first case, \( I_1 = 10 \, \text{A} \)
- Distance from the conductor, \( r = 20 \, \text{cm} = 0.2 \, \text{m} \) (convert cm to m for consistency in SI units)
- Magnetic field in the first case, \( B_1 = 50 \, \text{T} \)

### Step 2: Write the equation for the first case
Using the formula for the magnetic field, we can express \( B_1 \) as:
\[ B_1 = \frac{K \cdot I_1}{r} \]
Substituting the known values:
\[ 50 = \frac{K \cdot 10}{0.2} \]

### Step 3: Solve for \( K \)
Rearranging the equation to solve for \( K \):
\[ K = \frac{50 \cdot 0.2}{10} \]
\[ K = \frac{10}{10} = 1 \]

### Step 4: Write the equation for the second case
Now, we need to find the magnetic field \( B_2 \) for the second case where the current \( I_2 = 40 \, \text{A} \) and the distance \( r \) remains the same:
\[ B_2 = \frac{K \cdot I_2}{r} \]
Substituting the value of \( K \) and \( I_2 \):
\[ B_2 = \frac{1 \cdot 40}{0.2} \]

### Step 5: Calculate \( B_2 \)
Now, calculate \( B_2 \):
\[ B_2 = \frac{40}{0.2} = 200 \, \text{T} \]

### Final Answer
The value of the magnetic field for a 40 A current at the same distance is:
\[ B_2 = 200 \, \text{T} \]

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