Which of the following can have negative value ?

A. `chi`
B. `mu_(r)`
C. `mu_(0)`
D. `mu_(r)` and `chi` both

Options

A. `chi`
B. `mu_(r)`
C. `mu_(0)`
D. `mu_(r)` and `chi` both

Answer

To solve the question regarding which of the given options can have a negative value, let's analyze each option step by step.

### Step 1: Understanding the Terms
- **Chi (χ)**: This is the magnetic susceptibility of a material. It indicates how much a material will become magnetized in an applied magnetic field.
- **μ_r (Relative Permeability)**: This is the ratio of the permeability of a material (μ) to the permeability of free space (μ₀). It indicates how much more or less a material can be magnetized compared to free space.
- **μ₀ (Permeability of Free Space)**: This is a constant value representing the permeability of a vacuum, approximately equal to \(4\pi \times 10^{-7} \, \text{N/A}^2\).

### Step 2: Analyzing Each Option
- **Option A: χ (Chi)**
- The susceptibility (χ) can be negative for diamagnetic materials, which do not retain magnetic properties in the absence of an external magnetic field. Therefore, χ can have a negative value.

- **Option B: μ_r (Relative Permeability)**
- The relative permeability (μ_r) is defined as μ/μ₀. For diamagnetic materials, μ is less than μ₀, which makes μ_r less than 1. In some cases, μ_r can also be negative for certain materials. Therefore, μ_r can have a negative value.

- **Option C: μ₀ (Permeability of Free Space)**
- The permeability of free space (μ₀) is a constant and cannot be negative. Its value is always positive.

- **Option D: μ_r and χ both**
- Since both χ and μ_r can be negative, this option is also valid.

### Conclusion
Based on the analysis:
- **Option A (χ)** can be negative.
- **Option B (μ_r)** can also be negative.
- **Option C (μ₀)** cannot be negative.
- **Option D (μ_r and χ both)** is correct since both can be negative.

Thus, the correct answer is **Option D: μ_r and χ both** can have negative values.