The current density across a cylinder of radius 2mm the curret density varies with distance in x-axis according to j =103e-400r.find total current

To find the total current flowing through a cylinder where the current density varies with distance along the x-axis, we can use the formula for current density, which is defined as the current per unit area. In this case, the current density \( j \) is given by the equation \( j = 10^3 e^{-400r} \), where \( r \) is the radial distance from the center of the cylinder. The radius of the cylinder is 2 mm, which we need to convert to meters for consistency in units.

Understanding the Variables

First, let's clarify the variables involved:

• j: Current density in amperes per square meter (A/m²).
• r: Radial distance from the center of the cylinder in meters.
• R: The radius of the cylinder, which is 2 mm or 0.002 m.

Calculating Total Current

The total current \( I \) can be found by integrating the current density over the cross-sectional area of the cylinder. The formula for current is:

I = ∫ j(r) dA

In cylindrical coordinates, the differential area element \( dA \) can be expressed as:

dA = 2πr dr

Thus, the total current can be calculated as:

I = ∫ from 0 to R (j(r) * 2πr dr)

Setting Up the Integral

Substituting the expression for \( j \) into the integral gives:

I = ∫ from 0 to 0.002 (10^3 e^{-400r} * 2πr dr)

This simplifies to:

I = 2000π ∫ from 0 to 0.002 (r e^{-400r} dr)

Evaluating the Integral

To evaluate the integral \( ∫ r e^{-400r} dr \), we can use integration by parts. Let:

• u = r (which means du = dr)
• dv = e^{-400r} dr (which means v = -\frac{1}{400} e^{-400r})

Using integration by parts, we have:

∫ u dv = uv - ∫ v du

Substituting our values:

∫ r e^{-400r} dr = -\frac{1}{400} r e^{-400r} + \frac{1}{400} ∫ e^{-400r} dr

The integral of \( e^{-400r} \) is:

∫ e^{-400r} dr = -\frac{1}{400} e^{-400r}

Putting it all together, we get:

∫ r e^{-400r} dr = -\frac{1}{400} r e^{-400r} - \frac{1}{160000} e^{-400r}

Evaluating the Limits

Now we need to evaluate this from 0 to 0.002:

At r = 0.002:

- \frac{1}{400} (0.002) e^{-80} - \frac{1}{160000} e^{-80}

At r = 0:

0 (since both terms vanish).

Thus, the total current becomes:

I = 2000π \left[-\frac{1}{400} (0.002) e^{-80} - \frac{1}{160000} e^{-80}\right]

Final Calculation

Now, we can factor out \( e^{-80} \) and calculate the numerical values:

I = 2000π e^{-80} \left[-\frac{0.002}{400} - \frac{1}{160000}\right]

Calculating the coefficients gives:

I = 2000π e^{-80} \left[-0.000005 - 0.00000625\right]

I = 2000π e^{-80} \left[-0.00001125\right]

Finally, compute the total current using a calculator for \( e^{-80} \) and π to get the numerical value.

In summary, the total current flowing through the cylinder can be calculated by integrating the current density over the cross-sectional area, taking into account the exponential decay of the current density with respect to the radial distance. This method illustrates the application of calculus in solving problems related to current flow in cylindrical conductors.