Answer
a. The image is 30 cm from the mirror.
b. The image is real and inverted.
c. Ray tracing confirms the image is real and inverted at 30 cm from the mirror.
Solution
To solve the problem involving a concave mirror with a focal length of 18 cm and an object placed 45 cm from the mirror, we'll address each part step-by-step.
---
## **Given:**
- **Focal Length (f):** +18 cm (Positive for concave mirrors)
- **Object Distance (u):** -45 cm (According to the **Cartesian Sign Convention**, object distances are negative if they are in front of the mirror)
---
## **a. Determining the Image Distance (v)**
### **Mirror Equation:**
The relationship between the object distance (u), image distance (v), and focal length (f) is given by the mirror equation:
\[
\frac{1}{f} = \frac{1}{u} + \frac{1}{v}
\]
### **Plugging in the Known Values:**
\[
\frac{1}{18} = \frac{1}{-45} + \frac{1}{v}
\]
### **Solving for \( \frac{1}{v} \):**
\[
\frac{1}{v} = \frac{1}{18} - \frac{1}{45}
\]
\[
\frac{1}{v} = \frac{5 - 2}{90} = \frac{3}{90} = \frac{1}{30}
\]
### **Calculating \( v \):**
\[
v = +30 \text{ cm}
\]
### **Conclusion for Part a:**
- **Image Distance (v):** **+30 cm**
---
## **b. Nature of the Image (Real or Virtual, Upright or Inverted)**
### **Interpreting the Sign of \( v \):**
- **Positive \( v \):** Indicates that the image is formed on the same side as the object, which means the image is **real**.
- **Negative \( v \):** Would indicate a virtual image (not applicable here).
### **Orientation of the Image:**
- **Real Images:** Are typically **inverted** relative to the object.
- **Virtual Images:** Are typically **upright** relative to the object.
### **Conclusion for Part b:**
- **Image Nature:** **Real and Inverted**
---
## **c. Ray Tracing to Confirm Numerical Results**
Ray tracing involves drawing lines (rays) from the object to the mirror and determining where they converge to form the image. For concave mirrors, three principal rays are typically used:
1. **Parallel Ray:**
- **Path:** A ray parallel to the principal axis reflects through the **focal point (F)**.
- **Outcome:** Demonstrates that parallel rays converge at the focal point after reflection.
2. **Central Ray (Through the Center of Curvature):**
- **Path:** A ray passing through the **center of curvature (C)** reflects back along the same path.
- **Outcome:** Shows that rays passing through \( C \) retrace their path, confirming the location of the center of curvature.
3. **Focal Ray:**
- **Path:** A ray passing through the **focal point (F)** before hitting the mirror reflects parallel to the principal axis.
- **Outcome:** Ensures that rays emanating from \( F \) become parallel post-reflection.
### **Step-by-Step Ray Tracing:**
1. **Draw the Principal Axis:**
- Draw a straight horizontal line representing the principal axis.
- Mark the **mirror's surface** on one end.
2. **Mark the Focal Point:**
- From the mirror's surface, measure and mark the **focal point (F)** at +18 cm.
- The **center of curvature (C)** is at twice the focal length, so +36 cm from the mirror.
3. **Plot the Object:**
- Place the **object** 45 cm in front of the mirror along the principal axis.
4. **Draw the Three Rays from the Top of the Object:**
- **Parallel Ray:**
- Draw a ray from the top of the object parallel to the principal axis.
- After reflection, this ray passes through the focal point \( F \).
- **Central Ray:**
- Draw a ray from the top of the object passing through the center of curvature \( C \).
- This ray reflects back on itself.
- **Focal Ray:**
- Draw a ray from the top of the object passing through the focal point \( F \).
- After reflection, this ray becomes parallel to the principal axis.
5. **Locate the Image:**
- The point where the three reflected rays converge is the **image location**.
- According to our calculations, this should be at +30 cm from the mirror.
6. **Verify the Image Properties:**
- **Real Image:** The convergence of rays on the same side as the object confirms it's a real image.
- **Inverted Image:** The orientation of the image opposite to the object’s orientation confirms it's inverted.
### **Diagram Representation:**
While I cannot provide a visual diagram, the description above outlines how you would draw and analyze the rays to confirm that the image forms at 30 cm, is real, and is inverted.
---
## **Summary of Results:**
a. **Image Distance:** +30 cm from the mirror.
b. **Image Nature:** Real and Inverted.
c. **Ray Tracing Confirmation:** Using the three principal rays demonstrates that the image forms at 30 cm, is real, and inverted, aligning with the numerical calculations.
Reviewed and approved by the UpStudy tutoring team
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