---
title: "Value of n for rⁿ r solenoidal"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:value-n-r-r-solenoidal"
source: https://www.bctengineer.com/q/semester_2/mathematics/value-n-r-r-solenoidal
site: BCT Engineering
---

# Value of n for rⁿ r solenoidal

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2073 Magh, 5 marks; 2070 Magh, 5 marks (asked 2 times)
- Video: [For a position vector r = xi+yj+zk  |  Prove that - div(r^n r)  = ( n+3 ) r^n |  Bhagvati classes](https://www.youtube.com/watch?v=lXupd1cCzsU) — Bhagvati Classes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/value-n-r-r-solenoidal
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

Find the value of n so that rⁿ r is solenoidal.

## Short answer

The divergence of r^n r is (n+3) r^n. For a solenoidal field, divergence must be zero, so n+3=0, giving n = -3.

## Answer

A vector field F is solenoidal if ∇ · F = 0 everywhere (except possibly at the origin).

Let r = xi + yj + zk and r = |r| = √(x² + y² + z²), so ∂r/∂x = x/r, and so on.

#### Divergence of rⁿ r
Using ∇ · (φF) = ∇φ · F + φ(∇ · F) with φ = rⁿ and F = r:

```text
∇(rⁿ) = n r^(n−1) ∇r = n r^(n−1)(r/r) = n r^(n−2) r
∇ · (rⁿ r) = n r^(n−2) (r · r) + rⁿ (∇ · r)
           = n r^(n−2) r² + 3 rⁿ
           = (n + 3) rⁿ
```

#### Condition
(n + 3) rⁿ = 0 for all r ≠ 0 requires n + 3 = 0.

**Answer: n = −3.** The field r/r³ (the inverse-square field) is solenoidal.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A field F is solenoidal when:
   A. ∇ × F = 0
   B. ∇F = 0
   C. ∇ · F = 0
   D. F = 0
   Answer: C. Solenoidal means zero divergence. (Key point: Definition)

2. ∇(rⁿ) equals:
   A. n r^(n−1)
   B. n r^(n−2) r
   C. n rⁿ r
   D. r^(n−1) r/n
   Answer: B. ∇r = r/r so ∇(rⁿ) = n r^(n−1)(r/r). (Key point: Gradient of power)

3. ∇ · (rⁿ r) equals:
   A. (n + 3) rⁿ
   B. (n − 3) rⁿ
   C. n rⁿ
   D. 3 rⁿ
   Answer: A. n r^(n−2) r^2 + 3rⁿ. (Key point: Divergence)

4. The value of n is:
   A. 3
   B. 0
   C. −2
   D. −3
   Answer: D. (n+3) = 0; the field r/r^3 is solenoidal. (Key point: Answer)

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