Question

Two circular coils X (smaller) and Y (bigger), each having a single turn, carry equal currents in the same direction and subtend the same angle at point 'O' along the axis of the coil. The distances of the centers of coil to point 'O' are \(d\) and \((\frac{d}{2})\) for coil Y and X respectively. The radii of coils Y and X are \((2r)\) and \((r)\) respectively. The magnetic induction due to the bigger coil at point 'O' is \(B_y\) and that due to smaller coil X at point 'O' is \(B_x\). (\(d≫r\)) The relation between \(B_x\) and \(B_y\) is

• A. \(B_y = B_x\)
• B. \(B_y = 2B_x\)
• C. \(B_x = 2B_y\)
• D. \(B_x = 4B_y\)

Hint:

Use the axial field formula and the condition d much larger than r.

Answer 1

The Correct Option is C

Step 1: Scaling Argument:
Field of a distant coil goes as $R^2/x^3$. Going from Y to X: $R$ halves and $x$ halves.

Step 2: Factor:
$\dfrac{(1/2)^2}{(1/2)^3}=\dfrac{1/4}{1/8}=2$. So the smaller coil, being closer, gives twice the field.

Step 3: Answer:
$B_x=2B_y$. Option (C).

Final Answer:
Option (C). \[ \boxed{\text{(C) } B_x=2B_y} \]