Physics equations/18-Electric charge and field/Q:lineChargesCALCULUS

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A line charge on the y axis extends from y = -3 to y = 2. What is the y component of the electric field at the point (3, 7)?
Answer: ${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}\int _{\mathcal {A}}^{\mathcal {B}}{\frac {{\mathcal {C}}\;\lambda ds}{\left[{\mathcal {D}}^{2}+{\mathcal {E}}^{2}\right]^{\mathcal {F}}\;}}}$, where:

1

${\displaystyle {\mathcal {A}}=}$

2

${\displaystyle {\mathcal {B}}=}$

3

${\displaystyle {\mathcal {C}}=}$

4

${\displaystyle {\mathcal {D}}=}$

5

${\displaystyle {\mathcal {E}}=}$

6

${\displaystyle {\mathcal {F}}=}$

A line charge on the y axis extends from y = 2 to y = 7. What is the y component of the electric field at the point (2, 9)?
Answer: ${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}\int _{\mathcal {A}}^{\mathcal {B}}{\frac {{\mathcal {C}}\;\lambda ds}{\left[{\mathcal {D}}^{2}+{\mathcal {E}}^{2}\right]^{\mathcal {F}}\;}}}$, where:

1

${\displaystyle {\mathcal {A}}=}$

2

${\displaystyle {\mathcal {B}}=}$

3

${\displaystyle {\mathcal {C}}=}$

4

${\displaystyle {\mathcal {D}}=}$

5

${\displaystyle {\mathcal {E}}=}$

6

${\displaystyle {\mathcal {F}}=}$

A line charge on the x axis extends from x = 4 to x = 8. What is the y component of the electric field at the point (8, 4)?
Answer: ${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}\int _{\mathcal {A}}^{\mathcal {B}}{\frac {{\mathcal {C}}\;\lambda ds}{\left[{\mathcal {D}}^{2}+{\mathcal {E}}^{2}\right]^{\mathcal {F}}\;}}}$, where:

1

${\displaystyle {\mathcal {A}}=}$

2

${\displaystyle {\mathcal {B}}=}$

3

${\displaystyle {\mathcal {C}}=}$

4

${\displaystyle {\mathcal {D}}=}$

5

${\displaystyle {\mathcal {E}}=}$

6

${\displaystyle {\mathcal {F}}=}$

A line charge on the y axis extends from y = 4 to y = 6. What is the y component of the electric field at the point (5, 1)?
Answer: ${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}\int _{\mathcal {A}}^{\mathcal {B}}{\frac {{\mathcal {C}}\;\lambda ds}{\left[{\mathcal {D}}^{2}+{\mathcal {E}}^{2}\right]^{\mathcal {F}}\;}}}$, where:

1

${\displaystyle {\mathcal {A}}=}$

2

${\displaystyle {\mathcal {B}}=}$

3

${\displaystyle {\mathcal {C}}=}$

4

${\displaystyle {\mathcal {D}}=}$

5

${\displaystyle {\mathcal {E}}=}$

6

${\displaystyle {\mathcal {F}}=}$

A line charge on the x axis extends from x = 3 to x = 7. What is the x component of the electric field at the point (7, 8)?
Answer: ${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}\int _{\mathcal {A}}^{\mathcal {B}}{\frac {{\mathcal {C}}\;\lambda ds}{\left[{\mathcal {D}}^{2}+{\mathcal {E}}^{2}\right]^{\mathcal {F}}\;}}}$, where:

1

${\displaystyle {\mathcal {A}}=}$

2

${\displaystyle {\mathcal {B}}=}$

3

${\displaystyle {\mathcal {C}}=}$

4

${\displaystyle {\mathcal {D}}=}$

5

${\displaystyle {\mathcal {E}}=}$

6

${\displaystyle {\mathcal {F}}=}$