the total electric field e consists of the vector sum of two parts. one part has a magnitude of e1 = 1284 n/c and points at an angle θ1 = 30° above the +x axis. the other part has a magnitude of e2 = 1833 n/c and points at an angle θ2 = 62° above the +x axis. find the magnitude and direction of the total field. specify the directional angle relative to the +x axis.
Well, well, well, we have ourselves a puzzle to solve! Let's get cracking, shall we?
To find the magnitude of the total electric field E, we need to sum the two parts together. So, we have E = E1 + E2.
Using trigonometry, we can find the x and y components of each electric field vector. The x-component can be calculated as Ex = E * cos(θ), and the y-component can be calculated as Ey = E * sin(θ).
For E1, we have Ex1 = E1 * cos(θ1) and Ey1 = E1 * sin(θ1). Similarly, for E2, we have Ex2 = E2 * cos(θ2) and Ey2 = E2 * sin(θ2).
Now, let's add the x and y components of E1 and E2 together:
Ex = Ex1 + Ex2
Ey = Ey1 + Ey2
By using these equations, we can calculate the magnitudes of Ex and Ey.
Once we have Ex and Ey, the magnitude of the total electric field E can be found using the Pythagorean theorem: E = sqrt(Ex^2 + Ey^2).
Finally, to determine the direction of the total electric field, we can calculate the angle it makes with the +x axis using the inverse tangent function: θ = tan^(-1)(Ey/Ex).
Now, all you have to do is plug in the given values and crunch those numbers. Good luck, math wizard!
To find the magnitude and direction of the total electric field, we need to find the vector sum of the two components.
First, let's decompose the given electric fields into their x and y components:
For e1:
Magnitude, e1 = 1284 N/C
Angle, θ1 = 30° above the +x axis
The x-component of e1 can be found using trigonometry:
e1x = e1 * cos(θ1)
= 1284 * cos(30°)
≈ 1113 N/C
The y-component of e1 can be found using trigonometry:
e1y = e1 * sin(θ1)
= 1284 * sin(30°)
≈ 642 N/C
For e2:
Magnitude, e2 = 1833 N/C
Angle, θ2 = 62° above the +x axis
The x-component of e2 can be found using trigonometry:
e2x = e2 * cos(θ2)
= 1833 * cos(62°)
≈ 898 N/C
The y-component of e2 can be found using trigonometry:
e2y = e2 * sin(θ2)
= 1833 * sin(62°)
≈ 1605 N/C
Now, let's find the total electric field components:
The x-component of the total electric field is the sum of the x-components of e1 and e2:
etx = e1x + e2x
≈ 1113 N/C + 898 N/C
≈ 2011 N/C
The y-component of the total electric field is the sum of the y-components of e1 and e2:
ety = e1y + e2y
≈ 642 N/C + 1605 N/C
≈ 2247 N/C
Next, we can find the magnitude of the total electric field (et) using Pythagorean theorem:
et = sqrt(etx^2 + ety^2)
= sqrt((2011 N/C)^2 + (2247 N/C)^2)
≈ sqrt(4044121 + 5052009)
≈ sqrt(9096129)
≈ 3016 N/C
Finally, we can find the direction of the total electric field using trigonometry:
Directional angle, θt = atan(ety / etx)
= atan(2247 N/C / 2011 N/C)
≈ atan(1.116)
≈ 48.81°
Therefore, the magnitude of the total electric field is approximately 3016 N/C, and its directional angle relative to the +x axis is approximately 48.81°.
To find the magnitude and direction of the total electric field, we can use the concept of vector addition.
1. Start by finding the x and y components of each electric field:
- e1x = e1 * cos(θ1)
- e1y = e1 * sin(θ1)
- e2x = e2 * cos(θ2)
- e2y = e2 * sin(θ2)
In this case:
- e1x = 1284 N/C * cos(30°)
- e1y = 1284 N/C * sin(30°)
- e2x = 1833 N/C * cos(62°)
- e2y = 1833 N/C * sin(62°)
2. Add the x and y components separately:
- etx = e1x + e2x
- ety = e1y + e2y
3. Use the Pythagorean theorem to find the magnitude of the total electric field:
- et = sqrt(etx^2 + ety^2)
4. Use inverse trigonometric functions to find the direction angle of the total electric field relative to the +x axis:
- θt = arctan(ety / etx)
Now, let's perform the calculations:
1. Plug in the values and calculate the x and y components:
- e1x = 1284 N/C * cos(30°) = 1111.14 N/C
- e1y = 1284 N/C * sin(30°) = 642.00 N/C
- e2x = 1833 N/C * cos(62°) = 888.44 N/C
- e2y = 1833 N/C * sin(62°) = 1576.76 N/C
2. Add the x and y components separately:
- etx = 1111.14 N/C + 888.44 N/C = 1999.58 N/C
- ety = 642.00 N/C + 1576.76 N/C = 2218.76 N/C
3. Calculate the magnitude of the total electric field:
- et = sqrt((1999.58 N/C)^2 + (2218.76 N/C)^2) = 2897.13 N/C
4. Calculate the direction angle:
- θt = arctan(2218.76 N/C / 1999.58 N/C) = 49.79°
Therefore, the magnitude of the total electric field is 2897.13 N/C, and its direction angle relative to the +x axis is 49.79°.