In a certain region, `vec(B)` increases radially as `vec(E)=90 r(-hat(r))`. The magnitude of electric charge (in `mu C` ) contained within-a sphere of radius `2 m` centered at the origin is.

Answer

To find the magnitude of electric charge contained within a sphere of radius 2 m centered at the origin, we will follow these steps:

### Step 1: Understand the Electric Field
The electric field is given as:
\[ \vec{E} = 90r \hat{r} \]
This indicates that the electric field increases radially outward with a magnitude proportional to the distance \( r \).

### Step 2: Relate Electric Field to Charge Density
From Gauss's law, we know that the electric field \( \vec{E} \) is related to the charge density \( \rho \) by the equation:
\[ \vec{E} = \frac{\rho}{3 \epsilon_0} \hat{r} \]
where \( \epsilon_0 \) is the permittivity of free space.

### Step 3: Set Up the Equation
Equating the two expressions for the electric field:
\[ 90r \hat{r} = \frac{\rho}{3 \epsilon_0} \hat{r} \]
This implies:
\[ \rho = 270 \epsilon_0 \]

### Step 4: Calculate the Charge Density
Using the value of \( \epsilon_0 \):
\[ \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \]
Thus,
\[ \rho = 270 \times 8.85 \times 10^{-12} \approx 2.39 \times 10^{-9} \, \text{C/m}^3 \]

### Step 5: Calculate the Volume of the Sphere
The volume \( V \) of a sphere of radius \( r = 2 \, \text{m} \) is given by:
\[ V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi (8) = \frac{32}{3} \pi \, \text{m}^3 \]

### Step 6: Calculate the Total Charge
The total charge \( Q \) contained within the sphere can be calculated using:
\[ Q = \rho \cdot V \]
Substituting the values:
\[ Q = (2.39 \times 10^{-9} \, \text{C/m}^3) \cdot \left(\frac{32}{3} \pi \, \text{m}^3\right) \]
Calculating the volume:
\[ V \approx 33.51 \, \text{m}^3 \]
Thus,
\[ Q \approx 2.39 \times 10^{-9} \cdot 33.51 \approx 8.01 \times 10^{-8} \, \text{C} \]

### Step 7: Convert to Microcoulombs
To convert to microcoulombs:
\[ Q \approx 8.01 \times 10^{-8} \, \text{C} = 0.0801 \, \mu C \]

### Final Answer
The magnitude of electric charge contained within the sphere of radius 2 m is approximately:
\[ \boxed{0.08 \, \mu C} \]