Question

$PQR$ is a triangular frame made of a uniform metallic wire. $RS$ is a median of the triangular frame joining the vertex $R$ to the mid point $S$ of the wire $PQ$. (Wire $RS$ is also made of the same metallic wire). It is given that $PR = 6$ cm, $RQ = 8$ cm and $PQ = 10$ cm. The cross sectional area of the wire is $1$ $mm^2$ and its resistivity is $24 \times 10^{-9} \Omega m$. The triangle $PQR$ lies in a cylindrical region of magnetic field such that the intersection of the surface of the cylinder with the plane containing the frame forms the circum circle of triangle $PQR$. The magnetic field in the region decreases at the rate of $0.526$ T/s. The magnitude of the induced current in $mA$ in the median $RS$ of the frame is $5n$. Find $n$.

Answer 1

4.8

Answer 2

Here's a breakdown of the solution:

1. Identify the type of triangle $PQR$ using side lengths: it is a right-angled triangle at $R$.

2. Determine the circumcenter and circumradius: $S$ is the circumcenter, $RS$ is the median to the hypotenuse, $RS = PS = SQ = PQ/2 = 5$ cm.

3. Calculate the areas of triangles $PRS$ and $RQS$: each is half the area of triangle $PQR$.

4. Calculate the induced emf in loops $PRS$ and $RQS$ using Faraday's law and the rate of change of magnetic field.

5. Calculate the resistance of each segment of the wire using resistivity, cross-sectional area, and length.

6. Set up a circuit diagram with loops $PRS$ and $RQS$ and apply Kirchhoff's Voltage Law, considering the induced emfs and resistances.

7. Solve the system of linear equations for the loop currents $i_1$ and $i_2$.

8. Calculate the current in the median $RS$ as the sum of the loop currents.

9. Convert the current to mA and equate it to $5n$ to find the value of $n$.

The calculated current in RS is $0.024$ A $= 24$ mA. Given that the magnitude of the induced current in mA is $5n$. $24 = 5n$ $n = \frac{24}{5} = 4.8$.