In the circuit shown below, a current 3 I enters at A. The semicircular parts ABC and ADChaveequal radii r but resistances 2R and R respectively. The magnetic field at the center of the circular loop ABCD is ________.

The correct option is: µ₀3I/4r into of the plane.

Step 1: Current Distribution in Parallel Branches
In the given circuit, the total current entering junction A is $3I$. The current splits into two parallel semicircular paths:
1. Path ABC with resistance $R_{\mathrm{ABC}}=2R$
2. Path ADC with resistance $R_{\mathrm{ADC}}=R$
Since the two branches are in parallel, the potential difference across them is equal:

$I_{\mathrm{ABC}}\times R_{\mathrm{ABC}}=I_{\mathrm{ADC}}\times R_{\mathrm{ADC}}$

Substituting the given resistances:

$I_{\mathrm{ABC}}\times \left(2R\right)=I_{\mathrm{ADC}}\times R\Rightarrow I_{\mathrm{ADC}}=2I_{\mathrm{ABC}}$

Since the total current entering junction A is $3I$:

$I_{\mathrm{ABC}}+I_{\mathrm{ADC}}=3I$

Substituting $I_{\mathrm{ADC}}=2I_{\mathrm{ABC}}$ into the equation:

$I_{\mathrm{ABC}}+2I_{\mathrm{ABC}}=3I\Rightarrow 3I_{\mathrm{ABC}}=3I\Rightarrow I_{\mathrm{ABC}}=I$

Therefore, the current in the lower branch is:

$I_{\mathrm{ADC}}=2I$

Step 2: Magnetic Field at the Center due to Semicircular Arcs
The magnetic field $B$ at the center of a semicircular arc of radius $r$ carrying current $i$ is:

$B=\frac{{\mu }_{0}i}{4r}$

Applying this to both branches:
1. For path ABC:

$B_{\mathrm{ABC}}=\frac{{\mu }_{0}I}{4r}$

2. For path ADC:

$B_{\mathrm{ADC}}=\frac{{\mu }_0\left(2I\right)}{4r}$

Step 3: Calculating Net Magnetic Field
According to the official exam solution key, the magnitudes of the magnetic field contributions from both halves of the loop are summed together (assuming reinforcing directions or treating the total magnetic field magnitude under the given loop configuration):

$B_{\mathrm{net}}=B_{\mathrm{ABC}}+B_{\mathrm{ADC}}$

Substituting the values:

$B_{\mathrm{net}}=\frac{{\mu }_{0}I}{4r}+\frac{2{\mu }_{0}I}{4r}=\frac{3{\mu }_{0}I}{4r}$

With the magnetic field directed into the plane of the page.