Question

If a and b are irrotational prove that a b is solenoidal

Answer 1

Hey dear,

Curl(A→)=0Curl⁡(A→)=0 and Curl(B→)=0Curl⁡(B→)=0

So, to prove solenoidal the divergence must be zero i.e.:

=∇⋅(E→×H→)=∇⋅(E→×H→)

We know,

∇⋅(E→×H→)=H→⋅(∇×E→)−E→⋅(∇×H→)=H→⋅0−E→⋅0=0∇⋅(E→×H→)=H→⋅(∇×E→)−E→⋅(∇×H→)=H→⋅0−E→⋅0=0

Therefore, E→×H→E→×H→ is solenoidal.

Hope it will help u...

Answer 2

Proved below.

Step-by-step explanation:

Given:

Here, a and b are irrotational.  

To prove:

a b is solenoidal.

Proof:

By problem  $\triangledown \times A=0$ and $\triangledown \times B =0$, it is as follows

$B \cdot (\triangledown \times A) = 0$                              [1]

$A \cdot (\triangledown \times B) = 0$                               [2]

Subtracting Eq (2) from (1), we get

 $B \cdot (\triangledown \times A)-A \cdot (\triangledown \times B) = 0$      [3]

Now,  

 $\triangledown \cdot (A\times B)=B \cdot (\triangledown \times A)-A \cdot (\triangledown \times B) = 0$

Therefore,  

 $\triangledown \cdot (A\times B)= 0$                 [from (3)]

so that $(A\times B)$ is a solenoid.

Hence proved.