# Step-by-Step Solution Let's analyze the situation as described: - There are **two charges on the x-axis**. - We need to **find where the electric field is zero** on the x-axis: to the **left** of both charges, **between** them (middle), or to the **right**. ## Step 1: Assign Variables Let: - \( q_1 \) and \( q_2 \) be the two point charges. - \( q_1 \) is at \( x_1 \), \( q_2 \) at \( x_2 \), with \( x_1 < x_2 \). ## Step 2: Consider the Sign of Charges Assume the **common case**: - \( q_1 > \) (positive charge at \( x_1 \)) - \( q_2 < \) (negative charge at \( x_2 \)) ## Step 3: Analyze Each Region ### 1. **Left Region** (\( x < x_1 \)) - **Field due to \( q_1 \)**: Points away from \( q_1 \), i.e., to the left. - **Field due to \( q_2 \)**: Points toward \( q_2 \), which is to the right, so also to the left. - **Total Field**: Both fields **add** in the same direction (leftward). **Cannot cancel** each other. **No zero field here.** ### 2. **Middle Region** (\( x_1 < x < x_2 \)) - **Field due to \( q_1 \)**: Points away from \( q_1 \), to the right. - **Field due to \( q_2 \)**: Points toward \( q_2 \), also to the right. - **Total Field**: Both fields **add** (rightward). **Cannot cancel** each other. **No zero field here.** ### 3. **Right Region** (\( x > x_2 \)) - **Field due to \( q_1 \)**: To the right (away from \( q_1 \)). - **Field due to \( q_2 \)**: Toward \( q_2 \), which is to the left, so field points to the left. - **Total Field**: Fields are **opposite** in direction. **Possible to cancel** for some \( x > x_2 \). ## Step 4: Conclusion - **Only in the right region** (\( x > x_2 \)) do the fields due to the two charges point in **opposite directions**, so the net field can be zero somewhere here. ## Step 5: Why Not the Other Regions? - In the **left** and **middle** regions, the electric fields from both charges point in the **same direction** and therefore **cannot cancel**. --- # **Final Answer** ## The electric field can be zero **only in the right region** (to the right of both charges). **Reason:** Only in this region do the electric fields from the two charges point in opposite directions, making cancellation possible. In the left and middle regions, the fields point in the same direction and cannot cancel.