To calculate $I_{R_2}$ and $I_{R_3}$ in this circuit, we need to use the following steps based on circuit analysis principles (Kirchhoff's Laws and Ohm's Law):

1. Identify the parameters:

   • $R_1, R_2, R_3$: Resistances.
   • $E$: Voltage source.
   • $I$: Current from the battery.
   • $I_{R_2}$: Current through $R_2$.
   • $I_{R_3}$: Current through $R_3$.

2. Recognize the configuration:

   • $R_2$ and $R_3$ are in parallel.
   • The combination of $R_2$ and $R_3$ is in series with $R_1$.

3. Equivalent resistance of $R_2$ and $R_3$: The equivalent resistance of $R_2$ and $R_3$, denoted as $R_{23}$, is given by: $\frac{1}{R_{23}} = \frac{1}{R_2} + \frac{1}{R_3}$ or $R_{23} = \frac{R_2 R_3}{R_2 + R_3}.$

4. Total resistance in the circuit: The total resistance $R_{\text{total}}$ is: $R_{\text{total}} = R_1 + R_{23}.$

5. Find the total current $I$: Using Ohm's Law, the total current supplied by the source is: $I = \frac{E}{R_{\text{total}}}.$

6. Voltage across $R_2$ and $R_3$: The voltage $V_{R_{23}}$ across the parallel combination is: $V_{R_{23}} = I \cdot R_{23}.$

7. Currents through $R_2$ and $R_3$: Using Ohm's Law, the currents are: $I_{R_2} = \frac{V_{R_{23}}}{R_2},$ $I_{R_3} = \frac{V_{R_{23}}}{R_3}.$

If you provide the values for $E$, $R_1$, $R_2$, and $R_3$, I can calculate the specific values of $I_{R_2}$ and $I_{R_3}$. Would you like to proceed with this calculation?

Here are 5 related questions you might explore further:

1. How does changing $R_2$ or $R_3$ affect the current distribution?
2. What happens to $I$ if $R_1$ is increased?
3. Can we simplify the circuit further if $R_2$ equals $R_3$?
4. How would introducing another parallel resistor impact the equivalent resistance?
5. How does the power dissipated in $R_2$ compare to $R_3$?

Tip: When analyzing circuits, always start with equivalent resistances to simplify calculations.