Question

Two parallel wires carrying the currents I and 2I in the same direction have magnetic field B at the midpoint between them. If the current 2I is switched off, the magnetic field at that point will be:
A) 0.5B
B) B
C) 2B
D) 3B

Answer 1

Hint: The Biot – Sart law can be used to determine the magnetic field strength from the current segment. This is reduced to the simple case of an infinitely straight electrically conductive wire which is equal to $\text{B=}\dfrac{{{\text{ }\!\!\mu\!\!\text{ }}_{\text{0}}}\text{I}}{\text{2 }\!\!\pi\!\!\text{ R}}$.

Complete step by step solution:
The Biot-Savart Law can be used to determine the magnetic field strength from a current segment. Using this law-
Magnetic field at midpoint when 2I is on
$\begin{align}
  & \Rightarrow \dfrac{{{\text{ }\!\!\mu\!\!\text{ }}_{\text{0}}}\text{2I}}{\text{2 }\!\!\pi\!\!\text{ r}}+\dfrac{{{\text{ }\!\!\mu\!\!\text{ }}_{\text{0}}}\text{(-I)}}{\text{2 }\!\!\pi\!\!\text{ r}} \\
 & \Rightarrow \dfrac{{{\text{ }\!\!\mu\!\!\text{ }}_{\text{0}}}\text{I}}{\text{2 }\!\!\pi\!\!\text{ r}} \\
\end{align}$
Magnetic field at midpoint when 2I is switched off.
$\begin{align}
  & \Rightarrow \dfrac{{{\text{ }\!\!\mu\!\!\text{ }}_{\text{0}}}\text{0I}}{\text{2 }\!\!\pi\!\!\text{ r}}+\dfrac{{{\text{ }\!\!\mu\!\!\text{ }}_{\text{0}}}\text{(-I)}}{\text{2 }\!\!\pi\!\!\text{ r}} \\
 & \Rightarrow -\dfrac{{{\text{ }\!\!\mu\!\!\text{ }}_{\text{0}}}\text{I}}{\text{2 }\!\!\pi\!\!\text{ r}} \\
\end{align}$
So when switch is off magnitude will be same but in opposite direction

So option B is correct.

Additional Information: Magnetic field from current- The equation for the magnetic field strength (magnitude) produced by a long straight current-carrying wire is:
$\text{B=}\dfrac{{{\text{ }\!\!\mu\!\!\text{ }}_{\text{0}}}\text{I}}{\text{2 }\!\!\pi\!\!\text{ R}}$
For a long straight wire where I is the current, R is the shortest distance to the wire, and constant ${{\mu }_{0}}=4\pi \times {{10}^{-7}}\dfrac{m}{A}$ is the permeability of free space.Since the wire is very long, the magnitude of the field depends only on the distance from the wire r, not on the position along the wire. It is one of the simplest cases to calculate the magnetic field generated by a current.
The magnetic field of a long straight wire is more suspect than before. Each segment of the current produces a magnetic field like a long straight wire, and the total field of current of any size is the vector sum of the fields due to each segment. The formal description of the direction and magnitude of the field due to each section is called the Biot-Savart law. Integral algorithm is needed to sum the field to an arbitrary shape now. The Biot-Savart law is written in its complete form as $\int{\text{B}\text{.dl}=0}$, where the integral sums over the wire length where vector dℓ is the direction of the current.

Note: A more fundamental law than the Biot-Savart law is the Ampere Law, which deals with magnetic fields and current in a general way. It is written in integral form as $\int{\text{B}\text{.dl}=0}$ . Ampere's law: an equation that relates magnetic fields to the electric currents that cause them. Using Ampere's law, one can determine the magnetic field associated with a given current or current by connecting it to a given magnetic field, provided that the electric field does not change at the time.